PATENT ABSTRACT
Embodiments of cognitive radio technology can recover and utilize under-utilized portions of statically-allocated radio-frequency spectrum. A plurality of sensing methods can be employed. Transmission power control can be responsive to adjacent channel measurements. Digital pre-distortion techniques can enhance performance. Embodiments of a high DNR transceiver architecture can be employed.

PATENT DESCRIPTION
PRIORITY 
       [0001]    This application is related to and claims priority under 35 U.S.C. 119(e) to U.S. Provisional Patent Application No. 60/890,801 filed on Feb. 20, 2007 entitled “SYSTEM AND METHOD FOR COGNITIVE RADIO” by Haiyun Tang the complete content of which is hereby incorporated by reference. 
     
    
     BACKGROUND 
       [0002]    1. Field of the Invention 
         [0003]    The inventions herein described relate to systems and methods for cognitive radio. 
         [0004]    2. Description of the Related Art 
       Spectrum Utilization Problems 
       [0005]    A recent study by the FCC Spectrum Task Force [United States&#39; Federal Communications Commission (FCC), “Report of the spectrum efficiency working group,” November 2002, http://www.fcc.gov/sptf/files/IPWGFinalReport.pdf] found that while the available spectrum becomes increasingly scarce, the assigned spectrum is significantly underutilized. This imbalance between spectrum scarcity and spectrum underutilization is especially inappropriate in this Information Age, when a significant amount of spectrum is needed to provide ubiquitous wireless broadband connectivity, which is increasingly becoming an indispensable part of everyday life. 
         [0006]    Static spectrum allocation over time can also result in spectrum fragmentation. With lack of an overall plan, spectrum allocations in the US and other countries over the past several decades can appear to be random. 
         [0007]    Despite some efforts to serve best interests at the time, this leads to significant spectrum fragmentation over time. The problem is exacerbated at a global level due to a lack of coordinated regional spectrum assignments. In order to operate under such spectrum conditions, a device can benefit from operational flexibility in frequency and/or band shape; such properties can help to maximally exploit local spectrum availability. 
         [0008]    To address the above problems, an improved radio technology is needed that is capable of dynamically sensing and locating unused spectrum segments, and, communicating using these spectrum segments while essentially not causing harmful interference to designated users of the spectrum. Such a radio is generally referred to as a cognitive radio, although strictly speaking, it may perform only spectrum cognition functions and therefore can be a subtype of a broad-sense cognitive radio [J. M. III, “Cognitive radio for flexible mobile multimedia communications,”  Mobile Networks and Applications , vol. 6, September 2001.] that learns and reacts to its operating environment. Key aspects of a cognitive radio can include: 
         [0009]    Sensing: a capability to identify used and/or unused segments of spectrum. 
         [0010]    Flexibility: a capability to change operating frequency and/or band shape; this can be employed to fit into unused spectrum segments. 
         [0011]    Non-interference: a capability to avoid causing harmful interference to designated users of the spectrum. 
         [0012]    Such a cognitive radio technology can improve spectrum efficiency by dynamically exploiting underutilized spectrum, and, can operate at any geographic region without prior knowledge about local spectrum assignments. It has been an active research area recently. 
       FCC Spectrum Reform Initiatives 
       [0013]    FCC has been at the forefront of promoting new spectrum sharing technologies. In April 2002, the FCC issued an amendment to Part 15 rules that allows ultra-wideband (UWB) underlay in the existing spectrum [FCC, “FCC first report and order: Revision of part 15 of the commission&#39;s rules regarding ultra-wideband transmission systems,” ET Docket No. 98-153, April 2002]. In June 2002, the FCC established a Spectrum Policy Task Force (SPTF) whose study on the current spectrum usage concluded that “many portions of the radio spectrum are not in use for significant periods of time, and that spectrum use of these ‘white spaces’ (both temporal and geographic) can be increased significantly”. SPTF recommended policy changes to facilitate “opportunistic or dynamic use of existing bands.” In December 2003, FCC issued the notice of proposed rule making on “Facilitating Opportunities for Flexible, Efficient and Reliable Spectrum Use Employing Cognitive Radio Technologies” [FCC, “Facilitating opportunities for flexible, efficient, and reliable spectrum use employing cognitive radio technologies,” ET Docket No. 03-108, December 2003] stating that “by initiating this proceeding, we recognize the importance of new cognitive radio technologies, which are likely to become more prevalent over the next few years and which hold tremendous promise in helping to facilitate more effective and efficient access to spectrum.” 
         [0014]    While both UWB and cognitive radio are considered as spectrum sharing technologies, their approaches to spectrum sharing are substantially different. UWB is an underlay (below noise floor) spectrum sharing technology, while cognitive radio is an overlay (above noise floor) and interlay (between primary user signals) spectrum sharing technology as shown in  FIG. 1 . Through sensing combined with operational flexibility, a cognitive radio can identify and make use of spectral “white spaces” between primary user signals. Because a cognitive user signal resides in such “white spaces”, high signal transmission power can be permitted as long as signal power leakage into primary user bands does not embody harmful interference. 
         [0015]    Broadcast TV bands. 
         [0016]    Exemplary broadcast TV bands are shown in Graph  200  of  FIG. 2 . Each TV channel is 6 MHz wide. Between 0 and 800 MHz, there are a total of 67 TV channels (Channels 2 to 69 excluding Channel 37 which is reserved for radio astronomy). The NPRM [FCC, May 2004, op. cit.] excludes certain channels for unlicensed use: Channels 2-4, which are used by TV peripheral devices, and Channels 52-69, which are considered for future auction. Among the channels remaining, Channels 5-6, 7-13, 21-36, and 38-51 are available for unlicensed use in all areas. Unlicensed use in Channels 14-20 is allowed only in areas where they are not used by public safety agencies [FCC, May 2004, op. cit.]. 
         [0017]    It can be appreciated that Channels 52-69 are currently used by TV broadcasters and it is not clear if/when they will be vacated. There is significant interference in the lower channels 5-6 and 7-13. Based on these considerations, the spectrum segment 470-806 MHz covering TV channels 14-69 can be of particular interest. 
       Spectrum Opportunity in the TV Bands 
       [0018]    Spectrum opportunity can be a direct result of incumbent system inefficiency. In TV bands, a signal from a TV tower can cover an area with a radius of tens of kilometers. TV receivers can be sensitive to interference such that TV cell planning may be very conservative to ensure there is essentially no co-channel interference. This can leave a substantial amount of “white spaces” between co-channel TV cells as illustrated in the Map  300  of  FIG. 3 . Those “white spaces” can constitute an opportunistic region for cognitive users on a particular TV channel. Each TV channel may have a differently shaped opportunistic region. The total spectrum opportunity at any location can comprise the total number of opportunistic regions covering the location. A measurement in one locality shows an average spectrum opportunity in TV channels 14-69 of about 28 channels; that can be expressed as an equivalent bandwidth of approximately 170 MHz. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0019]      FIG. 1  graph of spectrum sharing technologies: UWB and cognitive radio 
           [0020]      FIG. 2  graph of exemplary television channel bands 
           [0021]      FIG. 3  map of television co-channel coverage areas and opportunistic region 
           [0022]      FIG. 4  diagram: cognitive radio system 
           [0023]      FIG. 5  diagram: amplification stages between antenna and ADC 
           [0024]      FIG. 6  diagram: heterodyne receiver 
           [0025]      FIG. 7  diagram: heterodyne transceiver 
           [0026]      FIG. 8  diagram: wideband direct-conversion receiver 
           [0027]      FIG. 9  graph: frequency-domain non-linear effect 
           [0028]      FIG. 10  diagram: double-ADC receiver architecture 
           [0029]      FIG. 11  diagram: double-ADC receiver architecture, detail 
           [0030]      FIG. 12  graph: image problem, image rejection filter 
           [0031]      FIG. 13  graph: solution for LO freq. with specified IF freq. 140 MHz 
           [0032]      FIG. 14  graph: solution for LO freq. with specified IF freq. 70 MHz 
           [0033]      FIG. 15  graph: solution for LO freq. with specified IF freq. 140 MHz and specified rejection margin 
           [0034]      FIG. 16  graph: example SAW filter response 
           [0035]      FIG. 17  graph: example SAW filter rejection mask 
           [0036]      FIG. 18  graph: RF gain requirements 
           [0037]      FIG. 19  diagram: heterodyne receiver, single-channel 
           [0038]      FIG. 20  diagram: wideband direct-conversion transmitter 
           [0039]      FIG. 21  graph: DTV transmission mask 
           [0040]      FIG. 22  Diagram: wideband direct-conversion transmitter, detail 
           [0041]      FIG. 23  graph: simulated signal spectra for specified device non-linearities. 
       
    
    
     DETAILED DESCRIPTION 
       [0042]      FIG. 4  depicts an embodiment of a cognitive radio system in block diagram. A transceiver  401  can be coupled with and/or in communication with one or more antennae  402 . Baseband signal processing can be provided by elements of a baseband processor  403 . Elements of a baseband processor  403  can comprise a sensing processor  404 , a transmit power control element  405 , and a pre-distortion element  406 . In some embodiments a pre-distortion element  406  can be coupled with and/or in communication with a transceiver  401 . In some embodiments a transmit power control element can be coupled with and/or in communication with a transceiver  401 . In some embodiments a collective sensing element  407  can be coupled with and/or in communication with a baseband processor  403  and/or elements comprising a baseband processor. 
         [0043]    In some embodiments transceiver  401  can comprise transceiver and/or transmitter and/or receiver mechanisms disclosed herein. In some embodiments sensing element  404  can comprise one or more sensing mechanisms as described herein. By way of example and not limitation these sensing mechanisms can include energy sensing, NTSC signal sensing, and/or ATSC signal sensing. In some embodiments a collective sensing element  407  can provide collective sensing mechanisms as described herein. 
         [0044]    In some embodiments transmit power control  405  can support adaptive transmit power control mechanisms described herein. In some embodiments pre-distortion element  406  can provide digital pre-distortion mechanisms as described herein. 
         [0045]    In some embodiments baseband processor  403  can support additional processing mechanisms as described herein. By way of example and not limitation these mechanisms can include filtering and/or reconstruction. 
       RF System Analysis 
     Input Signal Dynamic Range 
       [0046]    The diagram  200  of  FIG. 2  depicts an embodiment of a channel-based signal transmission scheme. Each of the channel signals in an embodiment can be considered to be independent. Hence, the total signal power over all channels considered (for example, TV Channels 14-69) can be computed as the sum of the individual signal powers of those channels. 
         [0047]    Considering the wideband signal over all the channels in an embodiment comprising TV channels, a total signal bandwidth can be 336 MHz and an antenna thermal noise floor over the signal bandwidth can be calculated: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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         [0048]    In some embodiments, a maximum measured signal power can be approximately −20 dBm. 
         [0049]    For an individual TV channel in an embodiment, a thermal noise floor can be 
         [0000]        n   0   dB =−174+10 log 10 (6×10 6 )≈−106 dBm  (2)
 
         [0050]    In some embodiments, a maximum single-channel power can have a value of approximately −20 dBm. In an embodiment of a cognitive radio system that operates close to the noise floor, a receiver can see a channel power disparity of 
         [0000]      −20−(−106+6)≈80 dB  (3)
 
         [0000]    assuming a receiver noise figure of 6 dB. 
       Third-Order Intermodulation 
       [0051]    In an ideal RF receive chain, all RF components can be perfectly linear and there is no distortion on the received signal after the signal has been processed by the RF receive chain. Real-world RF components—especially active RF components like amplifiers and mixers—can exhibit some degree of nonlinearity, resulting in signal distortion. Small-signal nonlinearity of a single RF component or cascaded RF components can be modeled by the following input-output relationship 
         [0000]        y ( t )=α 0 +α 1   x ( t )+α 2   x   2 ( t )+α 3   x   3 ( t )+  (4)
 
         [0000]    where x(t) is the input signal and y(t) is the output signal and in some typical embodiments the nonlinearity can be dominated by the low-order nonlinear terms. 
         [0052]    RF components typically operate on passband signals. For passband signals, even-order nonlinear terms can be discarded when appropriate filtering is performed on the RF chain. The small signal nonlinearity can then be approximated as: 
         [0000]        y ( t )≈α 1   x ( t )+α 3   x   3 ( t )  (5)
 
         [0000]    retaining only the lowest odd order distortion term. 
         [0053]    When a passband signal with baseband equivalent representation s B (t) passes through an element with nonlinear transfer function (3), the baseband equivalent representation of the output signal can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0054]    At the output, the ratio of the distortion power to the signal power, which is also the inverse of the dynamic range, can be expressed as: 
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         [0000]    is a factor that depends essentially only on the signal structure of s B (t). For example, Γ is approximately 7.5 dB if s B (t) is white noise.
 
Suppose s B (t) is a combined signal over all TV channels with power
 
         [0000]        P   In   =E[|s   B ( t )| 2 ]  (9)
 
         [0055]    The gain can be defined 
         [0000]        g=α   1   2   (10)
 
         [0000]    and output signal power 
         [0000]        P   Signal   =gP   In   (11)
 
         [0056]    Using a two-tone IP3 relationship 
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         [0057]    It can be appreciated that a third-order intercept point (IP3 or TOI) is the point at which a linear extrapolation (as a function of input power) of linear output power and third-order distortion power level meet. 
         [0000]    
       
         
           
             
               
                 
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         [0000]    Since the output 3rd-order distortion power is 
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         [0000]    then, according to Equations (7) and (14) 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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         [0058]    Note that the term I′ in Equation (16) accounts for added distortion that can result from a particular signal structure. When an input signal s B (t) is essentially a sinusoid (i.e. a single tone in frequency domain), Γ dB =0. 
       Overview of RF Receiver Functions 
       [0059]    The functions of a RF receiver system can comprise: a) Frequency translation and channel selection; and b) Signal amplification. 
       Direct RF Sampling 
       [0060]    An RF signal can reside in a particular frequency band 
         [0000]      [ f   c   −W,f   c   +W]   
         [0000]    where f c  is a carrier frequency and 2 W is a signal bandwidth. In order to retrieve information content from the signal, the signal can be digitized. 
         [0061]    In theory, it is possible to directly sample the RF signal at a carrier frequency. Such an approach, however, can be prohibitively expensive in terms of hardware cost and power consumption. For example, if a carrier frequency is 600 MHz, direct Nyquist sampling of an associated RF signal can require a sampling frequency at least 2(f c +W) or 1.2 GHz. In some embodiments an overall RF signal can contain both strong and weak signal contents, e.g. both TV signals and cognitive radio signals. A high-resolution ADC can be advantageously specified for some such embodiments. By way of non-limiting example, for a power difference between the strong and weak signals of 70 dB, an ADC with a resolution of at least 12 bits can be specified in some typical embodiments. Such ADC requirements can present realization challenges, given that some embodiments of current commercial ADCs can run at about 1 GHz sampling frequency, with 8-bit resolution [National Semiconductor Corporation, “ADC081000 High Performance, Low Power 8-Bit, 1 GSPS A/D Converter”, DS200681, 2004], [Maxim Integrated Products, “MAX108 Data Sheet: ±5V, 1.5 Gsps, 8-Bit ADC with On-Chip 2.2 GHz Track/Hold Amplifier”, 19-1492; Rev 1; 10/01]. Direct RF sampling embodiments may become a increasingly advantageous in the future, as ADC and related technologies evolve. 
       Frequency Translation and Channel Selection 
       [0062]    The high cost of RF direct sampling can be a result of the sampling of unnecessary signal contents below f c −W. Given an information bandwidth of 2 W, Nyquist sampling only requires a sampling frequency of 2 W in the circumstance that the signal center frequency can be shifted from the carrier frequency f c  to DC, i.e. 
         [0000]      [ f   c   −W,f   c   +W]→[−W,W]   (17)
 
         [0063]    Such frequency translation can typically be achieved in an RF receiver through mixing. In addition to performing frequency translation, a receiver can also perform channel selection in order to acquire a signal in the desired 2 W-wide information band. 
       Signal Amplification 
       [0064]    Another major function of an RF receiver can be signal amplification. Consider an 8-bit ADC receiving an input signal with peak-to-peak voltage of 600 mV [Nat&#39;l Semi. Corp., DS200681, 2004, op. cit.]. An associated quantization step can be 2.34 mV. The quantization noise power assuming a 50-Ohm load can be expressed 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    where a factor of 2 results from considering the total quantization noise power of the in-phase (I) and quadrature (Q) ADCs in the system. 
         [0065]    In some embodiments a received signal power level at the antenna can be small, e.g. close to the exemplary thermal noise level of −89 dBm in Equation (1). As illustrated in diagram  500  ( FIG. 5 ), significant amplification through multiple amplification stages along the RF chain can be provided in some embodiments to ensure that a signal has enough power to overcome a quantization noise floor when the signal reaches an ADC input. In some embodiments, a specification can be employed to ensure that quantization noise has a negligible impact on the system performance; require that at the ADC input, the total thermal noise (amplified thermal noise plus RF chain noise figure) is at least X dB  (e.g. 10 dB) above the quantization noise level. This specification can translate into a requirement on the total RF chain power gain g RF : 
         [0000]        g   RF   dB −89 dBm+ F   RF   dB ≧−47 dBm+ X   dB   (19)
 
         [0000]    where F RF  is the RF chain noise figure. Alternatively, this relationship can be expressed 
         [0000]        g   RF   dB ≧42 −F   RF   dB   +X   dB   (20)
 
         [0066]    As an example, consider a receiver with a noise figure of 6 dB and X dB =10 dB. The total gain provided by the RF chain needs to be at least 46 dB according to the above equation. Accomplishing this gain can be a non-trivial task. 
       Receiver Architecture Choices Based on Channel Selection Considerations 
       [0067]    Since each exemplary 6 MHz TV channel can carry dissimilar information content, in some embodiments channel selection can be employed to decode the information content of a particular channel, such as a TV channel. Channel selection can be performed at one or more of an RF stage, IF stage, analog baseband, digital baseband, and/or a combination of these stages. 
       RF Channel Selection: 
       [0068]    In one design scenario, a channel selection filter can be disposed in the RF stage immediately following the antenna in order to select the desired channel. Several problems can attend this approach. First, a high quality channel selection filter can present challenges to realization at specified RF frequencies. A quality metric for a filter can be defined as approximately its 3-dB bandwidth divided by its center frequency. For a specified fixed channel width, a corresponding quality metric value increases with increasing frequency. Hence, challenges to realizing such a filter can increase with frequency. In some embodiments a receiver can be specified to select any one of 55 TV channels from an exemplary TV band. Thus in some embodiments, a tunable RF channel selection filter can be employed, thereby further exacerbating realization challenges. In some application embodiments, a capability of simultaneous decoding multiple (eg., TV) channels can be specified. In some such embodiments a complete RF chain after a RF channel selection filter could be replicated for each additional channel, and can thereby increase cost and/or complexity of a realizable embodiment. 
       Heterodyne Receiver: 
       [0069]    Diagram  600  depicts a block diagram embodiment of a heterodyne receiver. 
         [0000]    Channel Selection in Some Embodiments of a Conventional Heterodyne Receiver can be Achieved Through a Combination of Filtering Stages Along a RF (Radio Frequency) Chain, which are Herein Described: 
         [0070]    An RF filter  604 , also called a band selection filter. In some embodiments this can be an RF frequency filter connected directly to and/or coupled with an antenna  602 . An RF filter  604  can select a frequency band of interest, such as an entire exemplary TV band, and can reject signals outside the frequency band of interest, e.g. 900 MHz cellular signals. 
         [0071]    An Image rejection (IR) filter  612 . In some embodiments this filter can be disposed prior to a RF mixer  614  in order to reject one or more image signals. In some embodiments an image signal can otherwise fold into a desired signal band after mixing [B. Razavi,  RF Microelectronics . Pearson-Prentice Hall, 1998]. 
         [0072]    An IF filter  616 , also called a channel selection filter. In some embodiments this filter can be primarily responsible for channel selection. In some embodiments an IF filter  616  can be realized as a standalone component, e.g. a surface acoustic wave (SAW) filter [C. Marshall and et al., “2.7 v GSM transceiver ICs with on-chip filtering,”  ISSCC Digest of Technical Papers , pp. 148-149, February 1995]. 
         [0073]    One or more baseband filters  624   634 , also called anti-aliasing filters. A baseband filter can be disposed prior to an analog to digital converter (ADC) in order to reject alias signals that can result from sampling. Diagram  600  depicts baseband filter  624  employed in combination with ADC  628 , and baseband filter  634  employed in combination with ADC  638 , corresponding respectively to I and Q signal paths of a receiver embodiment. 
         [0074]    In some embodiments, with the exception of a band selection (RF) filter  604 , each of the filters just described can provide a degree of channel selection. In some embodiments a channel selection (IF) filter  616  can be capable of providing the largest contribution to selectivity. In some embodiments a heterodyne receiver architecture can be relatively complex and/or costly if multiple channels are to be decoded simultaneously. In some embodiments, an RF chain comprising the elements after the IR filter can be replicated for each additional channel in order to support simultaneous decoding of multiple channels. 
         [0075]    RF filter  604  can receive a signal from antenna  602 . RF filter  604  can provide a filtering function to a received signal. Low noise amplifier LNA  610  can be coupled with and receive a filtered signal from RF filter  601 . 
         [0076]    LNA  610  can provide a gain function with low noise to a received signal. IR filter  612  can be coupled with and receive a gain-modified signal from LNA  610 . IR filter  612  can provide a filtering function to a received signal. Oscillator LO 1    608  can provide a signal that can be a tone signal at a specified frequency. RF mixer  614  can be coupled with and receive a filtered signal from IR filter  612 . RF mixer  614  can be coupled with and receive a signal that can be a tone signal at a specified frequency from oscillator LO 1    608 . RF mixer  614  can provide a mixing function, providing a signal responsive to a combination of a signal received from IR filter  612  and a signal received from oscillator LO 1    608 . IF filter  616  can be coupled with and receive a signal from RF mixer  614 . IF filter  616  can provide a filtering function to a received signal. IF amp  618  can be coupled with and receive a filtered signal from IF filter  616 . IF amp  618  can provide a gain function to a received signal. 
         [0077]    Oscillator LO 2    609  can provide a signal that can be a tone signal at a specified frequency. Quad splitter  623  can provide a quadrature splitting function to a received signal, thereby providing an in-phase (I) and a quadrature (Q) signal. Quad splitter  623  can be coupled with and receive a signal from Oscillator LO 2    609 . IF mixer  622  can be coupled with and receive a signal of a first specified phase from Quad splitter  623 . IF mixer  622  can be coupled with and receive a gain-modified signal from IF amp  618 . IF mixer  622  can provide a mixing function, providing a signal responsive to a signal received from Quad splitter  623  and responsive to a signal received from IF amp  618 . Similarly, IF mixer  632  can provide a mixing function, providing a signal responsive to a signal of a second specified phase received from Quad splitter  623  and responsive to a signal received from IF amp  618 . Each of the baseband filters  624   634  can provide a filtering function to a corresponding received signal. Baseband filter  624  can be coupled with and receive a signal from IF mixer  622 . Baseband filter  634  can be coupled with and receive a signal from IF mixer  632 . Each of the variable gain amplifiers (VGA)  626   636  can provide a variable gain to a corresponding received signal. VGA  626  can be coupled with and receive a filtered signal from baseband filter  624 . VGA  636  can be coupled with and receive a filtered signal from baseband filter  634 . 
         [0078]    Each of the analog to digital converters (ADC)  628   628  can provide an analog to digital conversion function to a corresponding received analog signal. ADC  628  can be coupled with and receive a gain-modified signal from VGA  626 . ADC  638  can be coupled with and receive a gain-modified signal from VGA  636 . ADC  628  can provide a baseband digital output signal corresponding to the first specified phase (I). ADC  638  can provide a baseband digital output signal corresponding to the second specified phase (Q). 
         [0079]    It can be appreciated that in alternative embodiments of a heterodyne receiver  600  and in other receiver and transmitter embodiments herein described, various gain elements can be omitted and/or their functions realized by any known and/or convenient method of providing signal gain. 
       Heterodyne Transceiver: 
       [0080]    Diagram  700  depicts a block diagram embodiment of a heterodyne transceiver. An upper portion of diagram  700  corresponds directly to the heterodyne receiver  600  discussed herein. It can be appreciated that upon coupling antenna  702  to the receiver architecture through switch  706 , there can be essentially a one-to-one correspondence between elements of the receiver  600  and elements of the receiver portion of the transceiver diagram  700 . 
         [0081]    The signal chain and function of the elements therein correspond directly and respectively between [Antenna  602 , RF filter  604 , LO 1    608 , LO 2    609 , LNA  610 , IR filter  612 , RF mixer  614 , IF filter  616 , IF amp  618 , IF mixer  622 , Quad splitter  623 , Baseband filter  624 , VGA  626 , ADC  628 , IF mixer  632 , Baseband filter  634 , VGA  636 , ADC  638 ] and [Antenna  702 , RF filter  704 , LO 1    708 , LO 2    709 , LNA  710 , IR filter  712 , RF mixer  714 , IF filter  716 , IF amp  718 , IF mixer  722 , Quad splitter  723 , Baseband filter  724 , VGA  726 , ADC  728 , IF mixer  732 , Baseband filter  734 , VGA  736 , ADC  738 ]. 
         [0082]    The receiver portion of diagram  700  further comprises a Splitter  720  that couples elements with each other: IF amp  718 , IF mixer  722 , and IF mixer  732 . Corresponding elements IF amp  618 , IF mixer  622 , and IF mixer  632  can be similarly coupled in the embodiment of diagram  600 . 
         [0083]    In some embodiments the transmitter portion of diagram  700  can be advantageously realized using design analysis and/or frequencies and/or element specifications and/or particular elements in common with the receiver portion. In some embodiments elements RF filter  704 , LO 1    708 , and LO 2    709  can be used in common. 
         [0084]    In some embodiments, elements of the transmitter [IR filter  712 , RF mixer  714 , IF filter  716 , IF amp  718 , Splitter  720 , IF mixer  722 , Quad splitter  723 , Baseband filter  724 , VGA  726 , IF mixer  732 , Baseband filter  734 , VGA  736 ] can be substantially similar to the corresponding and respective elements of the receiver [IR filter  762 , RF mixer  764 , IF filter  766 , IF amp  768 , Splitter  770 , IF mixer  772 , Quad splitter  773 , Baseband filter  774 , VGA  776 , IF mixer  782 , Baseband filter  784 , VGA  786 ]. 
         [0085]    Each of the digital to analog converters DAC  778   788  can provide a digital to analog conversion function to a corresponding received digital signal, thereby providing corresponding converted corresponding analog signals. Baseband filters  774   784  can each provide a filter function to a corresponding received signal. Baseband filter  774  can be coupled with and receive an analog signal from DAC  778 . Baseband filter  784  can be coupled with and receive an analog signal form DAC  788 . 
         [0086]    Oscillator LO 2    709  can provide a signal that can be a tone signal at a specified frequency. Quad splitter  773  can provide a quadrature splitting function to a received signal, thereby providing an in-phase (I) and a quadrature (Q) signal. Quad splitter  773  can be coupled with and receive a signal from Oscillator LO 2    709 . IF mixer  772  can be coupled with and receive a signal of a first specified phase from Quad splitter  773 . IF mixer  772  can be coupled with and receive a filtered signal from baseband filter  774 . IF mixer  772  can provide a mixing function, providing a signal responsive to a signal received from Quad splitter  773  and responsive to a signal received from Baseband filter  774 . Similarly, IF mixer  782  can provide a mixing function, providing a signal responsive to a signal of a second specified phase received from Quad splitter  773  and responsive to a signal received from Baseband filter  784 . 
         [0087]    Combiner  770  can provide a combining function, providing a signal responsive to the combination of two received signals. Combiner  770  can be coupled with and receive a signal corresponding to a first specified phase from IF mixer  772 . Combiner  770  can be coupled with and receive a signal corresponding to a second specified phase from IF mixer  782 . IF amp  778  can provide a gain function to a received signal. IF amp can be coupled with and receive a combined signal from Combiner  770 . IF filter can provide a filter function to a received signal. IF filter can be coupled with and receive a gain-modified signal from IF amp  778 . 
         [0088]    LO 1    708  can provide a signal that can be a tone signal at a specified frequency. RF mixer  764  can be coupled with and receive a filtered signal from IF filter  766 . RF mixer  764  can be coupled with and receive a signal that can be a tone signal at a specified frequency from LO 1    708 . RF mixer  764  can provide a mixing function, providing a signal responsive to a combination of a signal received from IF filter  766  and a signal received from LO 1    708 . IR filter  762  can provide a filter function to a received signal. IR filter  762  can be coupled with and receive a mixed signal from RF mixer  764 . PA  760  can provide a power amplification function to a received signal. PA  760  can be coupled with and receive a filtered signal from IR filter  762 . RF filter  704  can provide a filter function to a received signal. RF filter can be coupled with and receive a signal from PA  760  via Switch  706 . Switch  706  can selectably couple PA  760  with RF filter  704 . Antenna  702  can provide an antenna transmission function to a power amplified signal received from PA  760 . 
       Wideband Direct-Conversion Receiver: 
       [0089]    From the above discussion, it can be appreciated that as long as the channel selection starts from a particular RF stage, in some embodiments the RF chain from that stage onward can be replicated for each additional channel. In some embodiments it can be advantageous to defer channel selection all the way until the digital baseband. Such an embodiment can comprise a receiver that is capable of simultaneously decoding all of the channels in one or more specified bands, such as all of the TV channels in depicted in the graph  200 . Two issues can be addressed in such a system. 
         [0090]    First, there can be a need to have fast and high-resolution sampling, because an ADC in such an embodiment sees an entire band of interest, such as a TV band (Channels 14-69) with 336 MHz of bandwidth. 
         [0091]    Second, because before channel selection, the overall signal consists of the signals from all the channels, some of which can be strong while some of which can be weak, RF component nonlinearities can cause signal intermodulations between one or more channels and thus degrade system performance for the weak channels. Linearity requirements on RF components constituting embodiments of such an architecture can thus be relatively stringent, especially on components disposed near to the ADC because such components can be specified to operate on relatively high power signals and/or amplified input signals. 
         [0092]    Current technology trends of digital scaling along with advances in high-speed ADCs can favor such an approach. An RF system design for embodiments of such a wideband direct-conversion receiver is herein described; diagram  800  depicts an embodiment. Such an architecture may be considered wideband because the RF receiver can operate on an entire band of interest, such as an entire TV band of 336 MHz bandwidth. In some embodiments, a system comprises a direct-conversion architecture wherein an RF signal can be directly down-converted to a baseband. 
         [0093]    RF filter  804  can receive a signal from antenna  802 . RF filter  804  can provide a filtering function to a received signal. Low noise amplifier LNA  806  can be coupled with and receive a filtered signal from RF filter  804 . LNA  806  can provide a gain function with low noise to a received signal. 
         [0094]    Oscillator LO  810  can provide a signal that can be a tone signal at a specified frequency. Quad splitter  808  can provide a quadrature splitting function to a received signal, thereby providing an in-phase (I) and a quadrature (Q) signal. Quad splitter  808  can be coupled with and receive a signal from LO  810 . Mixer  820  can be coupled with and receive a signal of a first specified phase from Quad splitter  808 . Mixer  820  can be coupled with and receive a gain-modified signal from LNA  806 . Mixer  820  can provide a mixing function, providing a signal responsive to a signal received from Quad splitter  808  and responsive to a signal received from LNA  806 . Similarly, Mixer  830  can provide a mixing function, providing a signal responsive to a signal of a second specified phase received from Quad splitter  808  and responsive to a signal received from LNA  806 . Each of the Baseband filters  822   832  can provide a filtering function to a corresponding received signal. 
         [0095]    Baseband filter  822  can be coupled with and receive a signal from Mixer  820 . Baseband filter  830  can be coupled with and receive a signal from Mixer  830 . Each of the variable gain amplifiers (VGA)  824   834  can provide a variable gain to a corresponding received signal. VGA  824  can be coupled with and receive a filtered signal from Baseband filter  822 . VGA  834  can be coupled with and receive a filtered signal from baseband filter  832 . 
         [0096]    Each of the analog to digital converters (ADC)  826   836  can provide an analog to digital conversion function to a corresponding received analog signal. ADC  826  can be coupled with and receive a gain-modified signal from VGA  824 . ADC  638  can be coupled with and receive a gain-modified signal from VGA  834 . ADC  826  can provide a baseband digital output signal corresponding to the first specified phase (I). ADC  836  can provide a baseband digital output signal corresponding to the second specified phase (Q). 
       Receiver Chain Frequency Planning 
     System Frequency Planning: 
       [0097]    Referring to the TV band diagram  200  in  FIG. 2 , consider a wideband direct-conversion receiver over the frequency range from 470 MHz to 806 MHz that can span TV channels 14-69. Since Channel 37 (608-614 MHz) is not used, the center frequency of Channel 37 can be employed as a direct-conversion carrier frequency, i.e. 
         [0000]        f   c =611 MHz  (21)
 
         [0098]    A Nyquist bandwidth can be specified of 
         [0000]      2 W=400 MHz  (22)
 
         [0000]    covering the RF signal frequencies from 411 MHz to 811 MHz. A number of alternative ADCs with 400 MHz sampling frequency and above can be used in an embodiment [National Semiconductor Corporation, ADC081000, 2004 op.cit.], [Maxim Integrated Products, MAX108, 10/01, op. cit.], [Analog Devices, Inc. “AD12401 Data Sheet, Rev A.”, D05649-0-4/06(A), May 2006]. 
       Frequency-Domain Effect of Second-Order Nonlinearity: 
       [0099]    Referring to a signal path of the receiver block diagram  800 : prior to the quadrature mixing stage comprising Mixer elements  820   830 , there can typically be a plurality of amplification stages, e.g. low noise amplifier (LNA) and/or amplification within the mixers. In some embodiments, device nonlinearities in such amplification stages can cause spectral contamination. In order to ensure that frequency planning is adequate in the presence of such spectral contamination, consider an RF signal 
         [0000]        s   c ( t )= r ( t ) cos [2 πf   c +θ( t )]  (23)
 
         [0000]    corresponding to a baseband signal 
         [0000]        s   B ( t )= r ( t ) e   jθ(t)   (24)
 
         [0000]    which is spectrally limited to [−W,W]. Taking into account device nonlinearity, the signal after the amplification stages can be expressed as 
         [0000]        y ( t )≈α 0 +α 1   r ( t )cos [2 πf   c   t +θ( t )]+α 2   r   2 ( t )cos 2 [2 πf   c   t +θ( t )]+α 3   r   3 ( t )cos 3 [2 πf   c +θ( t )]  (25)
 
         [0000]    where under the small-signal condition, only the second-order and third-order nonlinearities are retained. Third-order nonlinearity is neglected since in-band third-order interference is inevitable. However, to insure against in-band second-order interference, consider the second-order nonlinearity term 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0100]    Consider Fourier Transform Pairs 
         [0000]        s   B ( t )· s   B ( t )           S   B ( f )           S   B ( f )
 
         [0000]        s*   B ( t )· s*   B ( t )           S*   B (− f )           S*   B (− f )
 
         [0000]        s   B ( t )· s*   B ( t )           S   B ( f )           S*   B (− f )  (27)
 
         [0000]    and since S B (f) is spectrally limited to [−W,W], all the above signal products (in the immediately preceding equations) can be spectrally limited to [−2 W,2 W]. The graph  900  of  FIG. 9  illustrates the above nonlinear effect. It is clear from the illustration that as long as a carrier frequency f c  satisfies 
         [0000]        f   c ≧3 W  (28)
 
         [0000]    a signal can be essentially free of second-order in-band interference. In some embodiments this condition can be satisfied by frequency planning, i.e. 
         [0000]      611 MHz= f   c &gt;3 W=600 MHz  (29)
 
         [0000]    Other Issues with Direct-Conversion Architecture: 
         [0101]    Although some embodiments of a direct-conversion architecture do not suffer an image problem as can some embodiments of a heterodyne architecture, there can remain a number of challenges to a practical implementation [B. Razavi, op. cit.]. In some embodiments, LO self-mixing can create a DC offset. In some embodiments, analog baseband circuitry can add considerable flicker noise—also called 1/f noise, since noise power can be proportional to 1/f. In some embodiments I/Q mismatch can occur if the I and Q signal paths are not precisely balanced. Challenges of DC offset and flicker noise—which can prominent around DC—can be addressed in some emboddiments of an improved receiver architecture by using an empty 6 MHz signal channel, such as Channel 37 of an exemplary TV band, at DC. In some embodiments, I/Q mismatch can be compensated through digital calibration techniques. 
       Receiver Chain Gain Planning 
       [0102]    In light of frequency planning as discussed above, an ADC can be selected for an improved receiver embodiment. Consider using National Semiconductor&#39;s ADC 081000, an 8-bit 1 GHz ADC [Nat&#39;l Semi. Corp., DS200681, 2004, op. cit.], as previously mentioned. A receiver chain amplification calculation can be as discussed herein regards Signal Amplification, and employed for each 6 MHz TV channel. Assuming ADC operation at a 800 MHz sampling frequency, a quantization noise per TV channel can be expressed 
         [0000]        n   q   dB   =N   q   dB −10 log 10 (800/6)≈−68 dBm  (30)
 
         [0000]    where N q  is quantization noise power as calculated in Equation (18). In order to scale noise contributions, RF chain gain g RF  can be specified such that thermal noise exceeds the quantization noise at the ADC. In other words, 
         [0000]        n   0   dB   +F   RF   dB   +g   RF   dB   ≧n   q   dB   +X   dB   (31)
 
         [0000]    Again a noise figure can be specified F RF   dB =6 dB and a margin X dB =10 dB so that 
         [0000]        g   RF   dB   ≧n   q   dB +4 −n   0   dB =42 dB  (32)
 
         [0103]    An ADC can be operating at twice a specified sampling rate of 400 MHz; this can account for the discrepancy between the result shown here and that in discussion regards Signal Amplification. 
         [0104]    In some embodiments a receiver chain can provides 42 dB of amplification as just described. When operating with a maximum received signal power of −20 dBm, an amplified signal at an ADC can have a power level of 
         [0000]        P   Signal =22 dBm  (33)
 
         [0000]    Amplified thermal noise at the ADC can have a power level of −89+42+6=−41 dBm. In order to have third order intermodulation (IM3) power remain below thermal noise power, according to Equation (16), a required condition can be 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       3 
                        
                       
                           
                       
                        
                       
                         P 
                         Signal 
                         dB 
                       
                     
                     - 
                     
                       2 
                        
                       
                           
                       
                        
                       
                         P 
                         
                           IP 
                            
                           
                               
                           
                            
                           3 
                         
                         dB 
                       
                     
                     + 
                     
                       Γ 
                       dB 
                     
                   
                   = 
                   
                     
                       
                         P 
                         
                           IM 
                            
                           
                               
                           
                            
                           3 
                         
                       
                       &lt; 
                       
                         - 
                         41 
                       
                     
                     ⇒ 
                     
                       
                         P 
                         
                           IP 
                            
                           
                               
                           
                            
                           3 
                         
                         dB 
                       
                       &gt; 
                       
                         
                           1 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               3 
                                
                               
                                   
                               
                                
                               
                                 P 
                                 Signal 
                                 dB 
                               
                             
                             + 
                             
                               Γ 
                               dB 
                             
                             + 
                             41 
                           
                           ) 
                         
                       
                     
                     ⇒ 
                     
                       
                         P 
                         
                           IP 
                            
                           
                               
                           
                            
                           3 
                         
                         dB 
                       
                       &gt; 
                       
                         
                           1 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               3 
                               × 
                               22 
                             
                             + 
                             0 
                             + 
                             41 
                           
                           ) 
                         
                       
                     
                     ⇒ 
                     
                       
                         P 
                         
                           IP 
                            
                           
                               
                           
                            
                           3 
                         
                         dB 
                       
                       &gt; 
                       
                         53.5 
                          
                         
                             
                         
                          
                         dBm 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where for simplicity, it can be assumed that Γ dB =0. Such a high IP3 can be difficult to realize in an embodiment. 
         [0105]    Another potentially complicating design consideration can be that a specified ADC has an input digitizing range of input (maximum) peak-to-peak 0.6V. A maximum input signal power for the I and Q ADCs can be computed as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       10 
                        
                       
                           
                       
                        
                       
                         
                           log 
                           10 
                         
                          
                         
                           ( 
                           
                             2 
                             × 
                             
                               
                                 0.3 
                                 2 
                               
                               50 
                             
                             × 
                             
                               10 
                               3 
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       5.6 
                        
                       
                           
                       
                        
                       dBm 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
         [0000]    far smaller than the amplified signal power of 22 dBm.
 
A novel Double-ADC Receiver Architecture
 
         [0106]    Diagram  1100  depicts an embodiment in some detail comprising the double-ADC architecture of diagram  1000 , and that can address some issues discussed herein; particularly challenges to realization of an embodiment. Some notable blocks are represented in diagram  1000 . An Amplification Stage  1   1004  can comprise an LNA and/or optional additional amplifications. In one embodiment the total gain provided by this stage can be 15 dB (after 1-to-2 splitting) and a receiver chain noise figure up to this point can be 5 dB. Given an exemplary maximum receiver input signal power of −20 dBm, signal power at the output of this amplification stage can be −5 dBm. 
         [0107]    Thermal noise power at the output of this amplification stage can be −89+15+5=−69 dBm. In order to maintain an IM3 power below the thermal noise floor, a specified IP3 of Amplification Stage  1  must be larger than 
         [0000]    
       
         
           
             
               
                 - 
                 5 
               
               + 
               
                 
                   
                     - 
                     5 
                   
                   - 
                   
                     ( 
                     
                       - 
                       69 
                     
                     ) 
                   
                 
                 2 
               
             
             = 
             
               27 
                
               
                   
               
                
               
                 dBm 
                 . 
               
             
           
         
       
     
         [0000]    In some embodiments, a maximum component-wise IP3 in this amplification stage can be somewhat higher than 27 dBm in order to take into account losses through passive components, e.g. splitters and filters, in the stage. 
         [0108]    A signal arriving at analog to digital converter ADC 1   1006  can be representative of an input signal received by antenna  1002 . A representative input signal can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           Ω 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             x 
                             k 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                            
                           
                             j2π 
                              
                             
                                 
                             
                              
                             
                               f 
                               k 
                             
                              
                             t 
                           
                         
                       
                     
                     + 
                     
                       n 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where x k (t) and f k  are the baseband signal and frequency of a k th channel respectively. The signal after ADC 1   1006  sampling can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             y 
                             ′ 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 k 
                                 ∈ 
                                 Ω 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                   x 
                                   k 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   j2π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     f 
                                     k 
                                   
                                    
                                   t 
                                 
                               
                             
                           
                           + 
                           
                             n 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                           
                             q 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               k 
                               ∈ 
                               Ω 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               [ 
                               
                                 
                                   
                                     x 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     n 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     q 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                              
                             
                                
                               
                                 j2π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   f 
                                   k 
                                 
                                  
                                 t 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           
                             ∑ 
                             
                               k 
                               ∈ 
                               Ω 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               [ 
                               
                                 
                                   
                                     x 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     q 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                              
                             
                                
                               
                                 j2π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   f 
                                   k 
                                 
                                  
                                 t 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where q(t) is quantization noise; n k (t) and q k (t) are baseband equivalent thermal noise and quantization noise on Channel k; and in the approximation, thermal noise can be ignored because thermal noise power per channel can be approximately −106+15+5=−86 dBm; this can be far smaller than quantization noise power per channel, i.e. −68 dBm. The maximum input signal power to ADC 1   1006 , i.e. E[|y(t)| 2 ], can be approximately −20+15=−5 dBm, which can be smaller than a maximum allowable ADC input signal power of 5.6 dBm. 
         [0109]    In a baseband, digital filtering can be performed (by Digital Filtering element  1008 ) to select one or more specified channels. After filtering, a subset of the selected channels can be selected Λ ⊂ Ω whose SNRs exceed 25 dB. Element Digital Filtering  1008  can be adapted to provide this capability. A signal corresponding to the selected set of channels can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       y 
                       Λ 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         ∈ 
                         Λ 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         [ 
                         
                           
                             
                               x 
                               k 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                           
                             
                               q 
                               k 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ] 
                       
                        
                       
                          
                         
                           j2π 
                            
                           
                               
                           
                            
                           
                             f 
                             k 
                           
                            
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
         [0110]    A signal y Λ (t) can be shown as y H (t) in some figures herein; the “H” subscript indicating correspondence to relatively high power channels of an input signal. Two operations can be employed with this set of channels. First, this set of channels can be sent to a digital baseband processing unit  1020  for decoding, since they have adequate SNRs. Second, an analog waveform can be reconstructed corresponding to the signal y Λ (t) using a DAC  1010 . A reconstructed analog waveform can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       y 
                       Λ 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           Λ 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 x 
                                 k 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             + 
                             
                               
                                 q 
                                 k 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           ] 
                         
                          
                         
                            
                           
                             j2π 
                              
                             
                                 
                             
                              
                             
                               f 
                               k 
                             
                              
                             t 
                           
                         
                       
                     
                     + 
                     
                       p 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p(t) is quantization noise from the DAC  1010 . 
         [0111]    Subtracting a reconstructed waveform y Λ (t) from an original signal y(t) can yield: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       y 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     - 
                     
                       
                         y 
                         Λ 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             ( 
                             
                               Ω 
                               - 
                               Λ 
                             
                             ) 
                           
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             x 
                             k 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                            
                           
                             j2π 
                              
                             
                                 
                             
                              
                             
                               
                                 f 
                                 k 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           Λ 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             q 
                             k 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                            
                           
                             j2π 
                              
                             
                                 
                             
                              
                             
                               
                                 f 
                                 k 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       p 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     + 
                     
                       n 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and is depicted as a signal comprising y L (t) that can be provided by summing node  1012  in  FIG. 10 , wherein y L (t) corresponds to relatively low power channels of an input signal. 
         [0112]    Since the remaining channels belong to a set Ω−Λ, and these channels can have signal powers less than 25 dB above the an exemplary per channel quantization noise floor of −68 dBm, a maximum signal power per channel can be −68+25=−43 dBm. In an exemplary worst case, all of the channels can have signal powers at −43 dBm and Ω−Λ can comprise an exemplary complete set of 55 TV channels. A worst-case power of the signal y(t)−y Λ (t) then can be: 
         [0000]      −43+10 log 10 (55)≈−25 dBm  (41)
 
         [0113]    In order to provide a total RF chain amplification of 42 dB with the first-stage amplification already providing 15 dB gain, the second-stage amplification  1014  can be required to provide an additional 27 dB gain. In the above worst case example, a signal power at input of ADC 2   1016  (after second-stage amplification  1014 ) can be 2 dBm. Amplified thermal noise power at input of ADC 2   1016  can be −89+42+6=−41 dBm. To maintain an IM3 below the thermal noise floor, an IP3 of 
         [0000]    
       
         
           
             
               2 
               + 
               
                 
                   2 
                    
                   
                     ( 
                     41 
                     ) 
                   
                 
                 2 
               
             
             = 
             
               23.5 
                
               
                   
               
                
               dBm 
             
           
         
       
     
         [0000]    for second amplification stage  1014  can be specified. 
         [0114]    The summing node  1012  can provide a signal comprising specified relatively low-power bands and/or channels of a representative input signal but also comprising uncancelled residual signal attributed to specified relatively high-power bands and/or channels. Digital filtering  1018  can be adapted to substantially remove undesirable energy corresponding to specified bands and/or channels such as high-power channels corresponding to signal y H (t). Digital filtering  1018  can provide an advantageously filtered signal to digital baseband processing  1020 . In some embodiments digital baseband processing  1020  can further process and/or decode such an advantageously filtered signal and can provide one or more individual channel signals corresponding to y L (t). 
         [0115]    In order to prevent significant noise figure degradation, quantization noise p(t) added by DAC  1010  can be kept small in comparison to thermal noise n(t) in Equation (40). An exemplary DAC can provide up to 16-bit resolution at 500 MHz with an output peak-to-peak voltage swing of 1V. Examples of such DACs include Analog Devices AD9726 [Analog Devices, Inc., “AD9726 Data Sheet, Rev A”, D04540-0-11/05(A), November 2005] and Maxim MAX5888 [Maxim Integrated Products, “MAX5888 Data Sheet: 3.3V, 16-Bit, 500 Msps High Dynamic Performance DAC with Differential LVDS Inputs”, 19-2726; Rev 3; 12/03]. A quantization noise power for a 15-bit DAC can be expressed 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               
                                  
                                 
                                   p 
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                  
                               
                               2 
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           10 
                            
                           
                               
                           
                            
                           
                             
                               log 
                               10 
                             
                             [ 
                             
                               2 
                               × 
                               
                                 
                                   
                                     ( 
                                     
                                       1 
                                       / 
                                       
                                         2 
                                         15 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 12 
                               
                                
                               
                                 
                                   10 
                                   3 
                                 
                                 50 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           ≈ 
                             
                            
                           
                             
                               - 
                               85 
                             
                              
                             
                                 
                             
                              
                             dBm 
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which is less than a specified thermal noise floor of −89+15+5=−69 dBm. 
         [0116]    Diagram  1100  shows in some detail a RF block diagram of an example direct-conversion double-ADC receiver. Many suitable components for an exemplary embodiment are identified herein, by way of non-limiting example. 
         [0117]    The system of diagram  1100  comprises individual processing elements well known in the art and/or described herein. Each of these elements is generally identified herein with a name and/or abbreviation that corresponds to its well known and/or herein described function. Analog filters comprise BandPass  1108 , Lowpass 1   1124   1174 , and ReConstruction  1130   1180 . Digital filtering and/or other specified digital signal processing comprises Digital Filtering  1127   1177 . Gain modifying elements comprise low noise amplifiers LNA 1   1106  and LNA 2   1110 , automatic gain control AGC 1   1122   1172  and AGC 2   1134   1184 . Analog to digital converters comprise AD  1126   1176   1136   1186 . Digital to analog converters comprise DA  1128   1178 . 
         [0118]    Splitters comprise elements  1112  and  1116 . Mixers comprise elements  1120  and  1170 . Summing nodes comprise elements  1132  and  1182 . Delay compensation elements comprise Delay Comp.  1125   1175 . 
         [0119]    Delay elements comprise Phase Shift  1118 . 
         [0120]    LNA 1   1106  can be selectably coupled with Antenna  1102  via switch  1104 . When so coupled, LNA 1   1106  can receive a signal from Antenna  1102 . BandPass  1108  can be coupled with and receive a signal from LNA 1   1106 . LNA 2  can be coupled with and receive a signal from BandPass  1108 . 
         [0121]    Splitter  1112  can be coupled with and receive a signal from LNA 2   1110 . Mixer  1120  can be coupled with and receive a signal from Splitter  1112 . Mixer  1120  can be coupled with and receive a signal from Splitter  1116 . 
         [0122]    Mixer  1170  can be coupled with and receive a signal from Splitter  1112 . Mixer  1170  can be coupled with and receive a signal from PhaseShift  1118 . PhaseShift  1118  can be coupled with and receive a signal from Splitter  1116 . Splitter  1116  can be coupled with and receive a signal from an oscillator LO  1114 . 
         [0123]    AGC 1   1122  can be coupled with and receive a signal from Mixer  1120 . 
         [0124]    Lowpass 1   1124  can be coupled with and receive a signal from AGC 1   1122 . 
         [0125]    Delay Comp.  1125  can be coupled with and receive a signal from Lowpass 1   1124 . 
         [0126]    Summing node  1132  can be coupled with and receive a signal from Delay Comp.  1125 . 
         [0127]    AD  1126  can be coupled with and receive a signal from Lowpass 1   1124 . 
         [0128]    Digital Filtering  1127  can be coupled with and receive a signal from AD  1126 . 
         [0129]    DA  1128  can be coupled with and receive a signal from Digital Filtering  1127 . 
         [0130]    ReConstruction  1130  can be coupled with and receive a signal from DA  1128 . 
         [0131]    Summing node  1132  can be coupled with and receive a signal from ReConstruction  1130 . 
         [0132]    AGC 2   1134  can be coupled with and receive a signal from Summing node  1132 . 
         [0133]    AD  1136  can be coupled with and receive a signal from AGC 2   1134 . 
         [0134]    AD  1136  can provide a baseband in-phase component signal. 
         [0135]    AGC 1   1172  can be coupled with and receive a signal from Mixer  1170 . 
         [0136]    Lowpass 1   1174  can be coupled with and receive a signal from AGC 1   1172 . 
         [0137]    Delay Comp.  1175  can be coupled with and receive a signal from Lowpass 1   1174 . 
         [0138]    Summing node  1182  can be coupled with and receive a signal from Delay Comp.  1175 . 
         [0139]    AD  1176  can be coupled with and receive a signal from Lowpass 1   1174 . 
         [0140]    Digital Filtering  1177  can be coupled with and receive a signal from AD  1176 . 
         [0141]    DA  1178  can be coupled with and receive a signal from Digital Filtering  1177 . 
         [0142]    ReConstruction  1180  can be coupled with and receive a signal from DA  1178 . 
         [0143]    Summing node  1182  can be coupled with and receive a signal from ReConstruction  1180 . 
         [0144]    AGC 2   1184  can be coupled with and receive a signal from Summing node  1182 . 
         [0145]    AD  1186  can be coupled with and receive a signal from AGC 2   1184 . 
         [0146]    AD  1186  can provide a baseband quadrature component signal. 
         [0147]    Exemplary digital-analog conversion devices can be specified: National Semiconductor&#39;s ADC081000 [Nat&#39;l Semi. Corp., DS200681, 2004, op. cit.], an 8-bit 1 GHz ADC, and Analog Devices&#39; AD9726 [Analog Devices, Inc., D04540-0-11/05(A), November 2005, op. cit.], a 16-bit 600 MHz DAC. As shown in Diagram  1100 , a first amplification stage comprises LNAs, bandpass filters, splitters, mixers, variable gain amplifiers, and lowpass filters, with a total gain of 15 dB and a noise figure of approximately 5 dB. Exemplary system components and a cascaded gain analysis are shown in the following table. 
         [0148]    Note that because of losses due to the passive components, e.g. splitters and filters, in some embodiments one or more amplifiers can be needed in a first amplification stage. In some embodiments a second amplification stage can consist of variable gain amplifiers. An IP3 calculation for a second amplification stage can assume a maximum input signal power of −25 dBm, as discussed herein. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                   
               
               
                   
                   
                   
                   
                   
                 Max. 
                   
                   
               
               
                   
                   
                   
                 Output 
                   
                 output 
               
               
                   
                 Vendor: 
                 NF 
                 NF 
                 Gain 
                 power 
                 IP3 
                 DR 
               
               
                 Name 
                 Part 
                 (dB) 
                 (dB) 
                 (dB) 
                 (dBm) 
                 (dBm) 
                 (dB) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 LNA1 
                 Mini- 
                 3.5 
                 3.5 
                 12 
                 −8 
                 47 
                 110  
               
               
                   
                 Circuits: 
               
               
                   
                 HELA- 
               
               
                   
                 10B 
               
               
                 Bandpass 
                 TBD 
                 3 
                 3.6 
                 −3 
                 −11 
                 ∞ 
                 ∞ 
               
               
                 LNA2 
                 Mini- 
                 3.5 
                 3.9 
                 12 
                 1 
                 47 
                 92 
               
               
                   
                 Circuits: 
               
               
                   
                 HELA- 
               
               
                   
                 10B 
               
               
                 Splitter 
                 Mini- 
                 4 
                 3.9 
                 −4 
                 −3 
                 ∞ 
                 ∞ 
               
               
                   
                 Circuits: 
               
               
                   
                 ZFSC- 
               
               
                   
                 2-2 
               
               
                 Mixer 
                 Mini- 
                 8 
                 4.1 
                 −8 
                 −11 
                 30 
                 82 
               
               
                   
                 Circuits: 
               
               
                   
                 ZFY-2 
               
               
                 AGC1 
                 Linear 
                 7 
                 4.9 
                 14 
                 3 
                 47 
                 88 
               
               
                   
                 Tech: 
               
               
                   
                 LT5514 
               
               
                 Lowpass1 
                 TBD 
                 8 
                 4.9 
                 −8 
                 −5 
                 ∞ 
                 ∞ 
               
               
                 AGC2 
                 Linear 
                 7 
                 5.1 
                 27 
                 2 
                 47 
                 90 
               
               
                   
                 Tech: 
               
               
                   
                 LT5514 
               
               
                   
               
             
          
         
       
     
         [0149]    The above discussions and analysis show a wideband direct-conversion double-ADC receiver using exemplary hardware components can provide enabling system performance levels for embodiments of a TV-band cognitive radio system, and, can allow simultaneous decoding of essentially all of the TV channels in a designated spectrum. 
         [0150]    A conventional single-channel heterodyne receiver can be considered as a reference and a cost-effective alternative to the embodiments above. A heterodyne receiver can use progressive filtering in an analog domain in order to improve channel selectivity. Although such a receiver may not have the capability of simultaneous decoding of multiple channels, neither does it require high-speed ADCs. It can also be instructive to compare the single-channel performance of the heterodyne receiver with that of the wideband receiver. 
         [0151]    IP3 requirements for realizable embodiments of a double-ADC architecture can be relatively stringent. In some embodiments, the worst-case IM3 interference can be allowed to be higher than the thermal noise floor. 
         [0152]    Remaining interference can then be removed in a digital domain through distortion compensation techniques. 
       A Reference Heterodyne Receiver Design 
       [0153]    RF system design embodiments of a conventional single-channel heterodyne receiver can serve as a reference point and as an alternative to wideband direct-conversion receiver embodiments discussed herein. 
       Heterodyne Frequency Planning 
       [0154]    Frequency planning for a heterodyne receiver can present further design challenges than that of a direct-conversion receiver. For some embodiments of a heterodyne receiver, two frequency translations can be required, i.e. from RF to IF and from IF to baseband (although frequency translation between IF and baseband can be achieved in some embodiments employing direct IF sampling and/or digital frequency synthesis). One of the key design issues of a heterodyne receiver embodiment can be specification of an intermediate frequency (IF). 
       IF Filtering: 
       [0155]    As discussed herein regards Receiver Architecture Choices, a main purpose of an IF stage in a heterodyne receiver can be to provide channel selection filtering, because effective filtering can be more easily accomplished at a relatively low IF frequency than at a relatively high RF frequency. Availability of off-the-shelf IF filters can contribute to a practical selection and/or specification of an IF frequency. 
         [0156]    A surface acoustic wave (SAW) filter can be a typical choice for IF channel selection. Some embodiments of exemplary commercially available SAW filters can have specified center frequencies of 40 MHz, 70 MHz, and 140 MHz [16,17]. 
       Image Rejection: 
       [0157]    Referring to Diagram  600  of  FIG. 6 : Mixer  614  can be a second-order device, that is, a device that does not differentiate between positive and negative frequencies. Consequently, after mixing, a down-converted signal can contain both an intended signal and an image signal as illustrated in Diagram  1200  of  FIG. 12 . 
         [0158]    Mathematically, an intended signal can be represented as 
         [0000]        R   s ( t )cos [2 πf   c   t+φ   s ( t )]  (43)
 
         [0000]    which can be band-limited to [f c −W,f c +W]; an image signal can be represented as 
         [0000]        R   i ( t )cos [2 πf   i   t+φ   i ( t )]  (44)
 
         [0000]    and mixing can use a tone signal 
         [0000]      cos(2 πL   LO   t ) 
         [0159]    A mixing operation can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                         4 
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
         [0160]    A filtering operation [f c −f LO −W,f c −f LO +W] can be applied to the signal after mixing, whereupon the second and fourth term in the above expression can essentially vanish. However, for an image signal at 
         [0000]        f   i =2 f   LO   −f   c   (46)
 
         [0000]    the third term above can become 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       1 
                       2 
                     
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                         R 
                         i 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
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                             2 
                              
                             
                                 
                             
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                                
                               
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                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
         [0161]    In other words, this signal can be in the same band, i.e. [f c −f LO −W,f c −f LO +W], as an intended signal after mixing (first term). One way to resolve the problem is to apply an image rejection (IR) filter  1202  before mixing as shown in the graph  1200  of  FIG. 12  so that an image signal at 2f LO −f c  can be rejected before a signal enters a mixer. 
       Frequency Planning: 
       [0162]    For some embodiments, an intended signal can be in a specified band such as [470,806] MHz. An ideal image rejection filter can be a brick-wall filter around a specified band. Suppose such an ideal IR (image rejection) filter is used in an embodiment: essentially full pass in [470,806] MHz and essentially infinite rejection otherwise. f c , f LO , and f IF  can be the carrier, LO, and IF frequencies, respectively. To have image-free mixing in some embodiments, the following conditions must be essentially met 
         [0000]      2 f   LO   −f   c &lt;470 or 2 f   LO   −f   c &gt;806  (48)
 
         [0000]        f   c   −f   LO   =+f   IF  or  f   c   −f   LO   =−f   If   (49)
 
         [0000]    Since 2f LO −f c  is an image frequency, the first condition above can suggest that the image frequency must stay in a rejection band of an IR filter. The second condition can be expressed as |f c −f LO |=f IF , where the absolute value is due to the properties of a realizable signal mixer. 
         [0163]    Given IF frequency candidates of 40 MHz, 70 MHz, and 140 MHz, the three possible IF frequencies can be substituted in the above conditions and the systems solved for possible solutions. Solutions can be advantageously perceived graphically, as shown in graphs  1300 ,  1400 , and  1500 . 
         [0164]    Graph  1300  corresponds to a condition (f IF =140 MHz). Line A  1302  corresponds to (2f LO −f c =470). Line B  1304  corresponds to (2f LO −f c =806). Line C  1306  corresponds to (f c −f LO =140). Line D  1308  corresponds to (f c −f LO =−140). 
         [0165]    A portion of line C  1306  shown in a region below line A  1302  (corresponding to (2f LO −f c &lt;470)) can be part of a solution, and, a portion of line D  1308  shown in a region above line B  1304  (corresponding to (2f LO −f c &gt;806)) can also be part of a solution. By way of non-limiting example, f c =500 MHz is shown to be in Solution Region_ 1   1310  and with an f LO =360 MHz, an image is thereby at 220 MHz and within a rejection region of the IR filter. Since each solution region can cover a part of the input signal frequency range (e.g. Solution Region_ 1   1310  can cover 750 MHz and below and Solution Region_ 2   1312  can cover 543 MHz and above), both regions can be necessary for an embodiment comprising an entire exemplary input frequency range, i.e. [470,806] MHz. Thus the constraints of Graph  1300  can lead to a practical realization for single-stage image-free IF mixing in some embodiments. 
         [0166]    Graph  1400  corresponds to a condition (f IF =70 MHz). Line A  1402  corresponds to (2f LO −f c =470). Line B  1404  corresponds to (2f LO −f c =806). Line C  1406  corresponds to (f c −f LO =70). Line D  1308  corresponds to (f c −f LO =−70). 
         [0167]    Graph  1400  shows a Gap  1414  between solution regions  1410   1412 , corresponding to a region wherein image-free mixing can not occur in some embodiments. For the constraints corresponding to graph  1400 , some embodiments employing 70 MHz IF filters for single-stage image-free IF mixing can fail to provide a solution for an entire exemplary TV band [470,806] MHz. 
         [0168]    A similar analysis can show that some embodiments employing 40 MHz IF filters under such constraints can fail to provide a solution covering an entire exemplary TV band [470,806] MHz. 
         [0169]    The conditions for Graph  1300  and Graph  1400  correspond to an ideal brick-wall IR filter over the signal band. In practice, typical filter embodiments can have gradual edge roll-offs. Thus in some embodiments margins can be employed at IR filter edges in order to provide a specified level of image rejection. By way of non-limiting example, a 100-MHz margin can be added to each side of an IR filter in order to account for edge roll-offs. An image rejection region can then be 
         [0000]      2 f   LO   −f   c &lt;370 and 2 f   LO   −f   c &gt;906  (50)
 
         [0170]    Graph  1500  shows a solution for the conditions discussed. Line A  1502  corresponds to (2f LO −f c =370). Line B  1504  corresponds to (2f LO −f c =906). Line C  1506  corresponds to (f c −f LO =140). Line D  1508  corresponds to (f c −f LO =−140). 
         [0171]    An advantageous overlap between Solution Region_ 1   1510  and Solution Region_ 2   1512  can be relatively smaller than the overlap shown in Graph  1300 . By way of non-limiting example, 650 MHz can be a cutoff frequency. For exemplary TV channels 14 (center 473 MHz) to 43 (center 647 MHz), an LO frequency can be 
         [0000]        f   LO   =f   c −140  (51)
 
         [0000]    and for exemplary TV channels 44 (center 653 MHz) to 69 (center 803 MHz), an LO frequency can be 
         [0000]        f   LO   =f   c +140  (52)
 
       Gain Planning 
       [0172]    Graph  1600  of  FIG. 16  depicts the response of an exemplary SAW filter [Vectron International, “Surface Acoustic Wave (SAW) Products” http://www.vectron.com/products/saw/saw.htm]. The filter has a specified passband of approximately 6 MHz. The specified rejection for two 6 MHz channels adjacent to the pass band can be specified as at least 15 dB (due to the finite roll-offs at filter edges as shown in the figure). Specified rejection for the channels not adjacent to the pass band can be at least 50 dB. The filter has a specified insertion loss of 22.5 dB. 
         [0173]    A SAW filter channel rejection mask as shown in  FIG. 17  can be assumed. A target channel k has 0 dB rejection. Rejection for adjacent channels k±1 is specified as 15 dB. Rejection for all other channels is specified as 40 dB. Some embodiments of SAW filters are able to essentially meet the specified requirements of such a rejection mask. 
         [0174]    As discussed regards Receiver Chain Gain Planning, a per channel thermal noise floor n 0   dB  of −106 dBm can be specified, a per channel quantization noise floor n q   dB  of −68 dBm can be specified, and a receiver chain noise figure F thrmRF   dB  of 6 dB can be specified. A SNR degradation due to the quantization noise can be required to be 0.46 dB, corresponding to an X dB  value of 10 dB. A total RF chain amplification requirement can be obtained from the following SNR equation 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           SNR 
                           Final 
                           dB 
                         
                         = 
                           
                          
                         
                           10 
                            
                           
                               
                           
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                             log 
                             10 
                           
                            
                           
                             
                               
                                 g 
                                 RF 
                               
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                                 k 
                               
                             
                             
                               
                                 
                                   g 
                                   RF 
                                 
                                  
                                 
                                   n 
                                   0 
                                 
                                  
                                 
                                   F 
                                   RF 
                                 
                               
                               + 
                               
                                 n 
                                 q 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           min 
                            
                           
                             { 
                             
                               
                                 30 
                                  
                                 
                                     
                                 
                                  
                                 dB 
                               
                               , 
                               
                                 
                                   P 
                                   k 
                                   dB 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
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                                       0 
                                       dB 
                                     
                                     + 
                                     
                                       F 
                                       RF 
                                       dB 
                                     
                                   
                                   ) 
                                 
                                 - 
                                 0.46 
                               
                             
                             } 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where SNR Final   dB  is SNR measured at the baseband input; g RF  is total RF chain gain; and P k  is input (received) signal power of a target channel. It can be appreciated that the SNR ceiling can be set to 30 dB in order to meet specified performance levels. Graph  1800  of  FIG. 18  depicts a graphical solution to Equation (53). As shown in the figure, for high input power levels the gain required can be reduced as a result of a SNR ceiling at 30 dB. 
         [0175]    A total input signal power can be expressed 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     k 
                   
                   + 
                   
                     ( 
                     
                       
                         P 
                         
                           k 
                           - 
                           1 
                         
                       
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                           + 
                           1 
                         
                       
                     
                     ) 
                   
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                           ′ 
                         
                       
                     
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                       l 
                     
                   
                 
               
               
                 
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                   54 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Ω′ can be a whole channel set excluding channels k and k±1. Assuming a SAW filter rejection mask as shown in Diagram  1700 , after SAW filtering, a total signal power can be expressed: 
         [0000]    
       
         
           
             
               
                 
                   
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                     k 
                   
                   + 
                   
                     
                       10 
                       
                         - 
                         
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                         l 
                       
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and a total signal power at an ADC (after RF chain amplification) can be expressed: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
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                         RF 
                       
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                         indicates text missing or illegible when filed 
                       
                     
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
         [0176]    A condition can be imposed that the signal powers of the two adjacent channels satisfy 
         [0000]    
       
         
           
             
               
                 
                   
                     10 
                      
                     
                         
                     
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                       10 
                     
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                     A 
                     dB 
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where A dB  can be a maximum specified adjacent channel power differential, e.g. 40 dB. Without this condition, adjacent channel leakage could overwhelm a signal in a desired channel (e.g. referring to the DTV transmission mask in Diagram  2100 ). Assuming a maximum total input signal power of −20 dBm, signal power at the ADC can have an upper bound P Bound  such that: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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                               P 
                               k 
                             
                           
                           + 
                           
                             
                               10 
                               
                                 - 
                                 
                                   40 
                                   10 
                                 
                               
                             
                              
                             
                               10 
                               
                                 - 
                                 
                                   20 
                                   10 
                                 
                               
                             
                           
                         
                         ⌉ 
                       
                     
                   
                   = 
                   
                     P 
                     Bound 
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where in the second inequality A dB  can be specified as 40 dB and a constraint that Σ lεΩ′ P l  is less than −20 dBm can be employed. This upper bound is plotted in Diagram  1800 . 
         [0177]    According to Diagram  1800 , a maximum possible signal power at an ADC can be less than −3 dBm. A thermal noise power, shown as Final noise power in Diagram  1800 , at this point can be −58 dBm. An IP3 requirement for an amplifier in the signal chain just prior to an ADC can then be expressed 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       - 
                       3 
                     
                     + 
                     
                       
                         
                           - 
                           3 
                         
                         - 
                         
                           ( 
                           
                             - 
                             58 
                           
                           ) 
                         
                       
                       2 
                     
                   
                   = 
                   
                     24.5 
                      
                     
                         
                     
                      
                     dBm 
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
           
         
       
     
         [0178]    In some embodiments, a 140 MHz IF signal can be down-converted to a baseband using a conventional down-conversion approach as shown in Diagram  600 . Alternative embodiments can employ direct IF sampling with digital down-conversion. In some embodiments, ADCs with 400 MHz and/or greater sampling frequencies [8,9,12] can be used to perform direct IF sampling. 
         [0179]    In some embodiments, an LNA and a mixer can provide enough gain to overcome a SAW filter insertion loss, which can have a typical value of 20 dB. Exemplary low-loss SAW filters (with 10 dB insertion loss) are available [Integrated Device Technology, Inc., “Saw Filter Products”, http://www.idt.com/?id=3350]. Employing such SAW filters in some embodiments can contribute to relaxing a specified amplification requirement on an LNA and mixer. As shown in Diagram  1800 , an RF chain can be specified to provide an adjustable gain range of 60 dB, i.e. from −20 dB to +40 dB. In some embodiments an LNA and mixer can provide a switchable gain step of 20 dB. One or more amplifier(s) following a SAW filter can then provide an adjustable gain of between 0 and 40 dB. This gain can be combined with a 20 dB LNA-mixer gain step and can provide a specified 60 dB dynamic range. Automatic gain control (AGC) can be employed to ensure correct gain levels at an LNA and mixer and gain-adjustable amplifier(s), under the condition of varying input signal powers, in order to achieve optimal system performance. 
       Example System: 
       [0180]    Diagram  1900  depicts a block diagram embodiment of an example single-channel heterodyne receiver wherein exemplary cascaded SAW filters can be used to achieve a desired level of channel selectivity. 
         [0181]    Many exemplary processing components are identified. 
         [0182]    The system of diagram  1900  comprises individual processing elements well known in the art and/or described herein. Each of these elements is generally identified herein with a name and/or abbreviation that corresponds to its well known and/or herein described function. Analog filters comprise BandPass  1908  and Lowpass  1926 . Exemplary SAW filters comprise IF Filter 1   1920  and IF Filter 2   1922 . Gain modifying elements comprise low noise amplifiers LNA 1   1906  and LNA 2   1910 , automatic gain control AGC 1   1918  and AGC 2   1924 . Analog to digital converters comprise AD  1928 . Mixers comprise Mixer  1916 . Attenuators comprise Attenuator  1912 . 
         [0183]    LNA 1   1906  can be selectably coupled with Antenna  1902  via switch  1904 . When so coupled, LNA 1   1906  can receive a signal from Antenna  1902 . BandPass  1908  can be coupled with and receive a signal from LNA 1   1906 . LNA 2   1910  can be coupled with and receive a signal from BandPass  1908 . Attenuator  1912  can be coupled with and receive a signal from LNA 2   1910 . Mixer  1916  can be coupled with and receive a signal from Attenuator  1912 . Mixer  1916  can be coupled with and receive a signal from Buffer  1914 . 
         [0184]    Buffer  1914  can provide an LO signal, as from an oscillator. 
         [0185]    AGC 1   1918  can be coupled with and receive a signal from Mixer  1916 . IF Filter 1   1920  can be coupled with and receive a signal from AGC 1   1918 . IF Filter 2   1922  can be coupled with and receive a signal from IF Filter 1   1920 . AGC 2   1924  can be coupled with and receive a signal from IF Filter 2   1922 . Lowpass  1926  can be coupled with and receive a signal from AGC 2   1924 . AD  1928  can be coupled with and receive a signal from Lowpass  1926 . 
         [0186]    AD  1186  can provide a baseband component signal. 
         [0187]    In some embodiments, an exemplary ADC, Analog Devices&#39; AD12401 [Analog Devices, Inc. AD12401, May 2006, op.cit.], a 12-bit 400 MHz ADC, can be used for direct IF sampling. The following table shows a system gain analysis. An exemplary SAW filter can have adjacent channel rejection of 8 dB and “Max. output power” can be reduced accordingly at the output of each SAW filter. Thus for some exemplary embodiments, a resulting overall system noise figure can be computed to be about 5.2 dB. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                   
               
               
                   
                   
                   
                   
                   
                 Max. 
                   
                   
               
               
                   
                   
                   
                 Output 
                   
                 output 
               
               
                   
                 Vendor: 
                 NF 
                 NF 
                 Gain 
                 power 
                 IP3 
                 DR 
               
               
                 Name 
                 Part 
                 (db) 
                 (dB) 
                 (dB) 
                 (dBm) 
                 (dBm) 
                 (dB) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 LNA1 
                 Mini- 
                 3.5 
                 3.5 
                 12 
                 −8 
                 47 
                 110  
               
               
                   
                 Circuits: 
               
               
                   
                 HELA- 
               
               
                   
                 10B 
               
               
                 Bandpass 
                 TBD 
                 3 
                 3.6 
                 −3 
                 −11 
                 ∞ 
                 ∞ 
               
               
                 LNA2 
                 Mini- 
                 3.5 
                 3.9 
                 12 
                 1 
                 47 
                 92 
               
               
                   
                 Circuits: 
               
               
                   
                 HELA- 
               
               
                   
                 10B 
               
               
                 Attenuator 
                 TBD 
                 4 
                 3.9 
                 −4 
                 −3 
                 ∞ 
                 ∞ 
               
               
                 Mixer 
                 Mini- 
                 8 
                 4.1 
                 −8 
                 −11 
                 30 
                 82 
               
               
                   
                 Circuits: 
               
               
                   
                 ZFY-2 
               
               
                 AGC1 
                 Linear 
                 7 
                 4.9 
                 17 
                 6 
                 47 
                 82 
               
               
                   
                 Tech: 
               
               
                   
                 LT5514 
               
               
                 IF Filter 1 
                 Sawtek: 
                 6 
                 4.9 
                 −6 
                 −8 
                 ∞ 
                 ∞ 
               
               
                   
                 854913 
               
               
                 IF Filter 2 
                 Sawtek: 
                 6 
                 4.9 
                 −6 
                 −22 
                 ∞ 
                 ∞ 
               
               
                   
                 854913 
               
               
                 AGC2 
                 Linear 
                 7 
                 5.2 
                 31 
                 9 
                 47 
                 76 
               
               
                   
                 Tech: 
               
               
                   
                 LT5514 
               
               
                 Lowpass 
                 TBD 
                 3 
                 5.2 
                 −3 
                 6 
                 ∞ 
                 ∞ 
               
               
                   
               
             
          
         
       
     
       Transmitter Architecture 
       [0188]    Diagram  2000  depicts an embodiment of a wideband direct-conversion transmitter comprising a similar structure as that of the wideband direct-conversion receiver of Diagram  800 . ADC elements  826   836  and DAC elements  2026   2036  have corresponding positions within the depicted signal processing chains, respectively. The position of LNA  806  corresponds to that of PA  2006 . Essentially the same frequency planning approaches as discussed regarding direct-conversion receiver embodiments can be employed regarding direct-conversion transmitter embodiments. In some embodiments, a mixing stage in diagram  2000  can perform an up-conversion function; the mixing stage can comprise Mixer  2020  and Mixer  2030 , and Quad splitter  2008 . 
         [0189]    Each of the digital to analog converters DAC  2026   2036  can provide a digital to analog conversion function to a corresponding received analog signal. 
         [0190]    Each of the converters DAC  2026   2036  can be provided with a baseband component signal (I and Q, respectively). 
         [0191]    Each of the Baseband filters  2022   2032  can provide a filtering function to a corresponding received signal. 
         [0192]    Baseband filter  2022  can be coupled with and receive a signal from DAC  2026 . Baseband filter  2032  can be coupled with and receive a signal from DAC  2036 . 
         [0193]    Oscillator LO  2010  can provide a signal that can be a tone signal at a specified frequency. 
         [0194]    Quad splitter  2008  can provide a quadrature splitting function to a received signal, thereby providing an in-phase (I) and a quadrature (Q) signal. 
         [0195]    Quad splitter  2008  can be coupled with and receive a signal from LO  2010 . 
         [0196]    Mixer  2020  can be coupled with and receive a signal of a first specified phase from Quad splitter  2008 . 
         [0197]    Mixer  2020  can be coupled with and receive a filtered signal from Baseband filter  2022 . 
         [0198]    Mixer  2030  can be coupled with and receive a signal of a second specified phase from Quad splitter  2008 . 
         [0199]    Mixer  2030  can be coupled with and receive a filtered signal from Baseband filter  2032 . 
         [0200]    Mixer  2020  can provide a mixing function, providing a signal responsive to a signal received from Quad splitter  2008  and responsive to a signal received from Baseband filter  2022 . 
       Similarly, 
       [0201]    Mixer  2030  can provide a mixing function, providing a signal responsive to a signal of a second specified phase received from Quad splitter  2008  and responsive to a signal received from Baseband filter  2032 . 
         [0202]    Tx Power Control  2007  can provide a transmission power control function to a received signal and/or received combination of signals. A transmission power control function can comprise a selectably adjustable gain and/or predistortion and/or any other known and/or convenient transmission power control techniques. 
         [0203]    Tx Power Control  2007  can be coupled with and receive a combination of signals from Mixer  2020  and Mixer  2030 . In some embodiments, a combiner element can be employed to combine signals from Mixer  2020  and Mixer  2030 . 
         [0204]    A power amplifier PA  2006  can provide a power amplification function to a received signal. 
         [0205]    PA  2006  can be coupled with and receive a signal from Tx Power Control  2007 . 
         [0206]    RF filter  2004  can provide a filtering function to a received signal. 
         [0207]    RF filter  2004  can be coupled with and receive a power-amplified signal from PA  2006 . 
         [0208]    Antenna  2002  can provide an antenna transmission function to a received signal. 
         [0209]    Antenna  2002  can be coupled with and receive a filtered signal from RF filter  2004 . 
         [0210]    Antenna  2002  can provide transmission of a signal responsive to a filtered signal received from RF filter  2004 . 
         [0211]    A maximum transmission power can be limited to 1 W or 30 dBm according to the NPRM [FCC, May 2004, op. cit.]. Considering the same exemplary 16-bit DAC as previously discussed, a maximum signal power out of the DAC can be calculated 
         [0000]    
       
         
           
             
               
                 
                   
                     10 
                      
                     
                         
                     
                      
                     
                       
                         log 
                         10 
                       
                        
                       
                         ( 
                         
                           2 
                           × 
                           
                             
                               0.5 
                               2 
                             
                             50 
                           
                           × 
                           
                             10 
                             3 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     10 
                      
                     
                         
                     
                      
                     dBm 
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
         [0212]    Alternative modulation schemes can have varying backoff requirements. For example, if OFDM is used, a backoff of 2.5 bits translating into a power loss of 15 dB can be required. A maximum signal power out of a DAC  2026   2036  can then be −5 dBm. A total transmitter RF chain amplification of 35 dB can then be needed before a signal reaches the antenna. A PA  2006  can typically provide 20 dB to 30 dB of gain. Additional amplification stages can then be needed between a PA  2006  and a DAC ( 2026  and/or  2036 ). 
         [0213]    Transmitter power control (TPC) can be helpful in improving wireless system capacity. TPC can be achieved using a variable gain amplifier  2007  as shown in Diagram  2000 . Alternatively, by employing a DAC with an ample number of bits (16), transmission power control can also be achieved using the DAC. For example, the top 8 bits of a DAC output can be dedicated to TPC. This can provide a total of 8×6=48 dB TPC range. In some embodiments, the remaining 8 DAC bits can be used for OFDM modulation: 2.5 bits for backoff and 5.5 bits for OFDM signal representation. 
         [0214]    The FCC may adopt the same DTV transmit mask as shown in Graph  200  for a TV-band cognitive radio. Given a modulation format, using the spectrum mask, linearity requirements of RF components can be derived. 
         [0215]    Since a PA can provide a last amplification stage, transmit chain nonlinearity can be dominated by that of the PA. Digital pre-distortion can be used for PA linearization. Digital pre-distortion techniques can be considered in a baseband system design. 
         [0216]    Diagram  2200  depicts a block diagram in some detail of an example embodiment of a wideband direct-conversion transmitter architecture essentially as depicted in Diagram  2000 . In some embodiments, an exemplary integrated wideband up-converter HMC497LP4 from Hittite Microwave can be used for signal up-conversion. In some embodiments, an exemplary Mini-Circuits ZHL-3010 amplifier can be used as a PA driver. In some embodiments, an Ophir 5303039A PA can have an output IP3 of 56 dBm and can provide an output power of 36 dBm with out-of-band emission level at −4 dBm. Notably, in some embodiments, every 1 dB reduction in transmission power can result in a 2 dB reduction in out-of-band emissions. 
         [0217]    Transmission power control can be employed in some embodiments to reduce out-of-band emissions. 
         [0218]    The system of diagram  2200  comprises individual processing elements well known in the art and/or described herein. Each of these elements is generally identified herein with a name and/or abbreviation that corresponds to its well known and/or herein described function. Analog filters comprise BandPass  2204  and Lowpass  2222   2232 . Gain modifying elements comprise Gain  2223   2233 , PA  2206 , and VGA  2207 . Digital to analog converters comprise DAC  2226   2236 . An Upconverter  2209  can comprise splitter/combiners, mixers, and a delay element. In some embodiments an Upconverter  2209  can be adapted to combine received (I) and (Q) baseband component signals into a signal having a modulating or carrier signal at the frequency of a received LO signal; hence “upconversion”. In some embodiments VGA  2207  can be adapted to provide transmission power control. 
         [0219]    Gain  2223  can be coupled with and receive a signal from DAC  2226 . Lowpass  2222  can be coupled with and receive a signal from Gain  2223 . Upconverter  2209  can be coupled with and receive a baseband component signal from Lowpass  2222 . Gain  2233  can be coupled with and receive a signal from DAC  2236 . Lowpass  2232  can be coupled with and receive a signal from Gain  2233 . Upconverter  2209  can receive an LO signal. 
         [0220]    VGA  2207  can be coupled with and receive a modulated signal from Upconverter  2209 . PA  2206  can be coupled with and receive a signal from VGA  2207 . BandPass  2204  can be coupled with and receive a signal from PA  2206 . Antenna  2202  can be selectably coupled via Switch  2203  with BandPass  2204 . When so coupled, Antenna  2202  can receive a signal from BandPass  2204  When so coupled, Antenna  2202  can provide transmission of a signal responsive to a filtered signal received from BandPass  2204 . 
       Baseband System Analysis: 
       [0221]    A baseband system design is described herein. 
         [0222]    FFT/IFFT-Based Digital Filtering and Reconstruction for Arbitrary Channel Rejection: 
         [0223]    A double-ADC architecture for a wideband direct-conversion TV-band cognitive radio receiver is herein described. An enabling function for this architecture can be channel rejection through digital filtering and reconstruction. Herein described is such a channel rejection method from a baseband perspective. 
         [0224]    Channel filtering can be accomplished using a common digital filter, e.g. a raised-cosine filter. It can also be achieved using an FFT and IFFT pair in combination. The latter approach can be especially efficient in simultaneous filtering of multiple channels, as required in some embodiments. 
         [0225]    Herein described are derivations of a continuous-time version of the operations of FFT/IFFT based filtering and reconstruction. Equivalent discrete-time version of the operations are subsequently described Channel Rejection Analysis: 
         [0000]    Referring to Equation (37), suppose a total signal is 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           ⋐ 
                           Ω 
                         
                       
                        
                       
                         
                           y 
                           k 
                         
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           k 
                           ⋐ 
                           Ω 
                         
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                 x 
                                 k 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             + 
                             
                               
                                 q 
                                 k 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           ] 
                         
                          
                         
                            
                           
                             j 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               f 
                               k 
                             
                              
                             t 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
         [0000]    from which a designated set of channels are to be rejected 
         [0000]    
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       l 
                       ∈ 
                       Λ 
                     
                   
                    
                   
                     
                       y 
                       l 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
           
         
       
     
         [0000]    An input signal can be truncated using a time-domain window w(t): 
         [0000]        y   1 ( t )= w ( t ) y ( t )  (63)
 
         [0000]    which can then be “FFT&#39;d” in order to generate a frequency-domain signal representation 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Y 
                       1 
                     
                      
                     
                       ( 
                       f 
                       ) 
                     
                   
                   = 
                   
                     
                       F 
                        
                       
                         [ 
                         
                           
                             y 
                             1 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         W 
                          
                         
                           ( 
                           f 
                           ) 
                         
                       
                       ⊗ 
                       
                         
                           ∑ 
                           
                             k 
                             ∈ 
                             Ω 
                           
                         
                          
                         
                           
                             Y 
                             k 
                           
                            
                           
                             ( 
                             f 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
           
         
       
     
         [0226]    To retrieve the signal on a particular channel lεΛ, a frequency-domain rectangular window on Y 1 (f) can be applied: 
         [0000]        Y   l ( f )=Π 2C ( f−f   l ) Y   1 ( f )  (65)
 
         [0000]    where Π 2C (f) is a rectangular window over the frequency range [−C,C] with 
         [0000]        C= 3 MHz+Δ  (66)
 
         [0000]    and Δ being the excess filter bandwidth. For all the channels in Λ, then 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Y 
                       ′ 
                     
                      
                     
                       ( 
                       f 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           l 
                           ∈ 
                           Λ 
                         
                       
                        
                       
                         
                           Y 
                           l 
                         
                          
                         
                           ( 
                           f 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             ∑ 
                             
                               l 
                               ∈ 
                               Λ 
                             
                           
                            
                           
                             
                               Π 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 C 
                               
                             
                              
                             
                               ( 
                               
                                 f 
                                 - 
                                 
                                   f 
                                   l 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                        
                       
                         
                           Y 
                           1 
                         
                          
                         
                           ( 
                           f 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
         [0227]    Note that for simplifying assumption that the channels in Λ are disjoint. In the case of contiguous channels, an overall rectangular window can be applied to the contiguous channels. The signal Y′(f) can then be transformed to time domain in order to generate y′(t) as a reconstructed version of the signals on the channels in Λ. 
         [0228]    In order to evaluate how much rejection can be achieved, the signal y′(t) can be subtracted from y 1 (t): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               y 
                               1 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               y 
                               ′ 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             
                               w 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                              
                             
                               y 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               y 
                               ′ 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               w 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             [ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   ∈ 
                                   
                                     ( 
                                     
                                       Ω 
                                       - 
                                       Λ 
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                 
                                   y 
                                   k 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             [ 
                             
                               
                                 
                                   w 
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       l 
                                       ∈ 
                                       Λ 
                                     
                                   
                                    
                                   
                                     
                                       y 
                                       l 
                                     
                                      
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                               
                               - 
                               
                                 
                                   y 
                                   ′ 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
         [0000]    So the remaining signal power on the channels in Λ can be expressed: 
         [0000]    
       
         
           
             
               
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     ∞ 
                   
                    
                   
                     
                       E 
                       [ 
                       
                         
                            
                           
                             
                               
                                 w 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     l 
                                     ∈ 
                                     Λ 
                                   
                                 
                                  
                                 
                                   
                                     y 
                                     l 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                             
                             - 
                             
                               
                                 y 
                                 ′ 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                            
                         
                         2 
                       
                       ] 
                     
                      
                     
                        
                       t 
                     
                   
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since a similar amount of rejection can be applied to any individual channel lεΛ, consider that Λ only contains one channel l as a simplifying assumption. Using Parseval&#39;s theorem 
         [0000]      ∫ −∞   ∞   E[|w ( t ) y   l ( t )− y   l ( t )| 2   ]dt=∫   −∞   ∞   E[|W ( f )           Y   l ( f )− Y   l ( f )| 2   ]df   (70)
 
         [0000]    Since the original signal power is 
         [0000]      ∫ −∞   ∞   E[|w ( t ) y   l ( t )| 2   ]dt=∫−∞   ∞   E[|W ( f )           Y   l ( f )| 2   ]df   (71)
 
         [0000]    rejection can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     R 
                     dB 
                   
                   = 
                   
                     10 
                      
                     
                       log 
                       10 
                     
                      
                     
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           ∞ 
                         
                          
                         
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                    
                                   
                                     
                                       W 
                                        
                                       
                                         ( 
                                         f 
                                         ) 
                                       
                                     
                                     ⊗ 
                                     
                                       
                                         Y 
                                         l 
                                       
                                        
                                       
                                         ( 
                                         f 
                                         ) 
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                               ] 
                             
                           
                            
                           
                              
                             f 
                           
                         
                       
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           ∞ 
                         
                          
                         
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                    
                                   
                                     
                                       
                                         W 
                                          
                                         
                                           ( 
                                           f 
                                           ) 
                                         
                                       
                                       ⊗ 
                                       
                                         
                                           Y 
                                           l 
                                         
                                          
                                         
                                           ( 
                                           f 
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         Y 
                                         l 
                                       
                                        
                                       
                                         ( 
                                         f 
                                         ) 
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                               ] 
                             
                           
                            
                           
                              
                             f 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   72 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Y l (f) can be assumed to be band-limited white Gaussian noise—a justified assumption according to the central limit theorem, if the signal x l (t) corresponds to filtered random data samples at 6 MHz, e.g. the DTV signal. This can result in 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         
                           
                             Y 
                             l 
                           
                            
                           
                             ( 
                             
                               f 
                               1 
                             
                             ) 
                           
                         
                          
                         
                           
                             Y 
                             l 
                             * 
                           
                            
                           
                             ( 
                             
                               f 
                               2 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               N 
                               0 
                             
                              
                             
                               δ 
                                
                               
                                 ( 
                                 
                                   
                                     f 
                                     1 
                                   
                                   - 
                                   
                                     f 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               f 
                               1 
                             
                             , 
                             
                               
                                 f 
                                 2 
                               
                               ∈ 
                               
                                 [ 
                                 
                                   
                                     
                                       f 
                                       l 
                                     
                                     - 
                                     B 
                                   
                                   , 
                                   
                                     
                                       f 
                                       l 
                                     
                                     + 
                                     B 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           Otherwise 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   73 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where in some embodiments B=3 MHz. A spectral power of the original signal, i.e. E[|W(f)         Y l (f)| 2 ], can be calculated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               
                                  
                                 
                                   
                                     W 
                                      
                                     
                                       ( 
                                       f 
                                       ) 
                                     
                                   
                                   ⊗ 
                                   
                                     
                                       Y 
                                       l 
                                     
                                      
                                     
                                       ( 
                                       f 
                                       ) 
                                     
                                   
                                 
                                  
                               
                               2 
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           E 
                            
                           
                             [ 
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   W 
                                    
                                   
                                     ( 
                                     
                                       f 
                                       - 
                                       u 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     Y 
                                     l 
                                   
                                    
                                   
                                     ( 
                                     u 
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   u 
                                 
                                  
                                 
                                   
                                     ∫ 
                                     
                                       - 
                                       ∞ 
                                     
                                     ∞ 
                                   
                                    
                                   
                                     
                                       
                                         W 
                                         * 
                                       
                                        
                                       
                                         ( 
                                         
                                           f 
                                           - 
                                           v 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         Y 
                                         l 
                                         * 
                                       
                                        
                                       
                                         ( 
                                         v 
                                         ) 
                                       
                                     
                                      
                                     
                                        
                                       v 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           E 
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∫ 
                                       
                                         
                                           f 
                                           l 
                                         
                                         - 
                                         B 
                                       
                                       
                                         
                                           f 
                                           l 
                                         
                                         + 
                                         B 
                                       
                                     
                                      
                                     
                                       
                                         W 
                                          
                                         
                                           ( 
                                           
                                             f 
                                             - 
                                             u 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           Y 
                                           l 
                                         
                                          
                                         
                                           ( 
                                           u 
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         u 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∫ 
                                       
                                         
                                           f 
                                           l 
                                         
                                         - 
                                         B 
                                       
                                       
                                         
                                           f 
                                           l 
                                         
                                         + 
                                         B 
                                       
                                     
                                      
                                     
                                       
                                         
                                           W 
                                           * 
                                         
                                          
                                         
                                           ( 
                                           
                                             f 
                                             - 
                                             v 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           Y 
                                           l 
                                           * 
                                         
                                          
                                         
                                           ( 
                                           v 
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         v 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∫ 
                             
                               
                                 f 
                                 l 
                               
                                
                               
                                   
                               
                                
                               B 
                             
                             
                               
                                 f 
                                 l 
                               
                               + 
                               B 
                             
                           
                            
                           
                             
                               ∫ 
                               
                                 
                                   f 
                                   l 
                                 
                                  
                                 
                                     
                                 
                                  
                                 B 
                               
                               
                                 
                                   f 
                                   l 
                                 
                                 + 
                                 B 
                               
                             
                              
                             
                               
                                  
                                 u 
                               
                                
                               
                                  
                                 
                                   vW 
                                    
                                   
                                     ( 
                                     
                                       f 
                                       - 
                                       u 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               W 
                               * 
                             
                              
                             
                               ( 
                               
                                 f 
                                 - 
                                 v 
                               
                               ) 
                             
                           
                            
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                   
                                     Y 
                                     l 
                                   
                                    
                                   
                                     ( 
                                     u 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     Y 
                                     l 
                                     * 
                                   
                                    
                                   
                                     ( 
                                     v 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∫ 
                             
                               
                                 f 
                                 l 
                               
                               - 
                               B 
                             
                             
                               
                                 f 
                                 l 
                               
                               + 
                               B 
                             
                           
                            
                           
                             
                               ∫ 
                               
                                 
                                   f 
                                   l 
                                 
                                 - 
                                 B 
                               
                               
                                 
                                   f 
                                   l 
                                 
                                 + 
                                 B 
                               
                             
                              
                             
                               
                                  
                                 u 
                               
                                
                               
                                  
                                 
                                   vW 
                                    
                                   
                                     ( 
                                     
                                       f 
                                       - 
                                       u 
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                 
                                   W 
                                   * 
                                 
                                  
                                 
                                   ( 
                                   
                                     f 
                                     - 
                                     v 
                                   
                                   ) 
                                 
                               
                                
                               
                                 N 
                                 0 
                               
                                
                               
                                 δ 
                                  
                                 
                                   ( 
                                   
                                     u 
                                     - 
                                     v 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             N 
                             0 
                           
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 B 
                               
                               
                                 + 
                                 B 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     W 
                                      
                                     
                                       ( 
                                       
                                         f 
                                         - 
                                         
                                           f 
                                           l 
                                         
                                         - 
                                         u 
                                       
                                       ) 
                                     
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                  
                                 u 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   74 
                   ) 
                 
               
             
           
         
       
     
         [0229]    Now considering the spectral power after rejection, i.e. E[|W(f)         Y l (f)−Y l (f)| 2 ]. Inner terms can be expressed: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               W 
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                             ⊗ 
                             
                               
                                 Y 
                                 l 
                               
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               Y 
                               l 
                             
                              
                             
                               ( 
                               f 
                               ) 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             
                               W 
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                             ⊗ 
                             
                               
                                 Y 
                                 l 
                               
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               
                                 Π 
                                 
                                   2 
                                    
                                   C 
                                 
                               
                                
                               
                                 ( 
                                 
                                   f 
                                   - 
                                   
                                     f 
                                     l 
                                   
                                 
                                 ) 
                               
                             
                             [ 
                             
                               
                                 W 
                                  
                                 
                                   ( 
                                   f 
                                   ) 
                                 
                               
                               ⊗ 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     ∈ 
                                     Ω 
                                   
                                 
                                  
                                 
                                   
                                     Y 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     f 
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           
                             
                               W 
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                             ⊗ 
                             
                               
                                 Y 
                                 l 
                               
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               
                                 Π 
                                 
                                   2 
                                    
                                   C 
                                 
                               
                                
                               
                                 ( 
                                 
                                   f 
                                   - 
                                   
                                     f 
                                     l 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   W 
                                    
                                   
                                     ( 
                                     f 
                                     ) 
                                   
                                 
                                 ⊗ 
                                 
                                   
                                     Y 
                                     l 
                                   
                                    
                                   
                                     ( 
                                     f 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               1 
                               - 
                               
                                 
                                   Π 
                                   
                                     2 
                                      
                                     C 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     f 
                                     - 
                                     
                                       f 
                                       l 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                            
                           
                             
                               W 
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                             ⊗ 
                             
                               
                                 Y 
                                 l 
                               
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   75 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where an approximation can be taken because the signal Y l (f) on channel l inside the rectangular window Π 2C (f−f l ) is far stronger (which is the reason it is being rejected) than the signals on the other channels whose power leakages into the channel are then negligible. From the above, it follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               
                                  
                                 
                                   
                                     
                                       W 
                                        
                                       
                                         ( 
                                         f 
                                         ) 
                                       
                                     
                                     ⊗ 
                                     
                                       
                                         Y 
                                         l 
                                       
                                        
                                       
                                         ( 
                                         f 
                                         ) 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       Y 
                                       l 
                                     
                                      
                                     
                                       ( 
                                       f 
                                       ) 
                                     
                                   
                                 
                                  
                               
                               2 
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     Π 
                                     
                                       2 
                                        
                                       C 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       f 
                                       - 
                                       
                                         f 
                                         l 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             2 
                           
                            
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                    
                                   
                                     
                                       W 
                                        
                                       
                                         ( 
                                         f 
                                         ) 
                                       
                                     
                                     ⊗ 
                                     
                                       
                                         Y 
                                         l 
                                       
                                        
                                       
                                         ( 
                                         f 
                                         ) 
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               1 
                               - 
                               
                                 
                                   Π 
                                   
                                     2 
                                      
                                     C 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     f 
                                     - 
                                     
                                       f 
                                       l 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                            
                           
                             N 
                             0 
                           
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 B 
                               
                               
                                 + 
                                 B 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     W 
                                      
                                     
                                       ( 
                                       
                                         f 
                                         - 
                                         
                                           f 
                                           l 
                                         
                                         - 
                                         u 
                                       
                                       ) 
                                     
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                  
                                 u 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   76 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   Let 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       K 
                        
                       
                         ( 
                         f 
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         
                           - 
                           B 
                         
                         
                           + 
                           B 
                         
                       
                        
                       
                         
                           
                              
                             
                               W 
                                
                               
                                 ( 
                                 
                                   f 
                                   - 
                                   u 
                                 
                                 ) 
                               
                             
                              
                           
                           2 
                         
                          
                         
                            
                           u 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   77 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The rejection can then be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           R 
                           = 
                             
                            
                           
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   N 
                                   0 
                                 
                                  
                                 
                                   K 
                                    
                                   
                                     ( 
                                     
                                       f 
                                       - 
                                       
                                         f 
                                         l 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   f 
                                 
                               
                             
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       
                                         Π 
                                         
                                           2 
                                            
                                           C 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           f 
                                           - 
                                           
                                             f 
                                             l 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                                  
                                 
                                   N 
                                   0 
                                 
                                  
                                 
                                   K 
                                    
                                   
                                     ( 
                                     
                                       f 
                                       - 
                                       
                                         f 
                                         l 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   f 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   K 
                                    
                                   
                                     ( 
                                     f 
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   f 
                                 
                               
                             
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       
                                         Π 
                                         
                                           2 
                                            
                                           C 
                                         
                                       
                                        
                                       
                                         ( 
                                         f 
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                                  
                                 
                                   K 
                                    
                                   
                                     ( 
                                     f 
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   f 
                                 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   or 
                 
               
               
                 
                   ( 
                   78 
                   ) 
                 
               
             
             
               
                 
                   
                     R 
                     dB 
                   
                   = 
                   
                     10 
                      
                     
                       
                         log 
                         10 
                       
                       [ 
                       
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             ∞ 
                           
                            
                           
                             
                               K 
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                              
                             
                                
                               f 
                             
                           
                         
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             ∞ 
                           
                            
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     Π 
                                     
                                       2 
                                        
                                       C 
                                     
                                   
                                    
                                   
                                     ( 
                                     f 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                              
                             
                               K 
                                
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                              
                             
                                
                               f 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   79 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Assuming that the time-domain window is a raised-cosine window: 
         [0000]    
       
         
           
             
               
                 
                   
                     w 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               1 
                               2 
                             
                             + 
                             
                               
                                 1 
                                 2 
                               
                                
                               cos 
                                
                               
                                 { 
                                 
                                   
                                     π 
                                     
                                       β 
                                        
                                       
                                           
                                       
                                        
                                       
                                         T 
                                         w 
                                       
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       t 
                                       + 
                                       
                                         
                                           1 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             1 
                                             - 
                                             β 
                                           
                                           ) 
                                         
                                          
                                         
                                           T 
                                           w 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 } 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   β 
                                 
                                 ) 
                               
                                
                               
                                 T 
                                 w 
                               
                             
                             ≤ 
                             t 
                             ≤ 
                             
                               
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   β 
                                 
                                 ) 
                               
                                
                               
                                 T 
                                 w 
                               
                             
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                             
                               
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   β 
                                 
                                 ) 
                               
                                
                               
                                 T 
                                 w 
                               
                             
                             ≤ 
                             t 
                             ≤ 
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   β 
                                 
                                 ) 
                               
                                
                               
                                 T 
                                 w 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               1 
                               2 
                             
                             + 
                             
                               
                                 1 
                                 2 
                               
                                
                               cos 
                                
                               
                                 { 
                                 
                                   
                                     π 
                                     
                                       β 
                                        
                                       
                                           
                                       
                                        
                                       
                                         T 
                                         w 
                                       
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       t 
                                       - 
                                       
                                         
                                           1 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             1 
                                             - 
                                             β 
                                           
                                           ) 
                                         
                                          
                                         
                                           T 
                                           w 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 } 
                               
                             
                           
                         
                         
                           
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   β 
                                 
                                 ) 
                               
                                
                               
                                 T 
                                 w 
                               
                             
                             ≤ 
                             t 
                             ≤ 
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   β 
                                 
                                 ) 
                               
                                
                               
                                 T 
                                 w 
                               
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           Otherwise 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with frequency-domain representation: 
         [0000]    
       
         
           
             
               
                 
                   
                     W 
                      
                     
                       ( 
                       f 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         sin 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           fT 
                           w 
                         
                       
                       
                         π 
                          
                         
                             
                         
                          
                         f 
                       
                     
                      
                     
                       
                         cos 
                          
                         
                           ( 
                           
                             πβ 
                              
                             
                                 
                             
                              
                             
                               fT 
                               w 
                             
                           
                           ) 
                         
                       
                       
                         1 
                         - 
                         
                           4 
                            
                           
                             β 
                             2 
                           
                            
                           
                             f 
                             2 
                           
                            
                           
                             T 
                             w 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
     
         [0230]    In some embodiments a further assumption can be employed that an FFT of size N is employed on input signal samples at 400 MHz such that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       T 
                       w 
                     
                      
                     
                       ( 
                       
                         1 
                         + 
                         β 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         N 
                          
                         
                           1 
                           400 
                         
                       
                       ⇒ 
                       
                         T 
                         w 
                       
                     
                     = 
                     
                       
                         N 
                         / 
                         400 
                       
                       
                         ( 
                         
                           1 
                           + 
                           β 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   82 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where T w  is expressed in μs. Note that in some embodiments the subcarrier spacing (inverse of the FFT period) can be: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       400 
                       N 
                     
                      
                     
                         
                     
                      
                     MHz 
                   
                   = 
                   
                     
                       
                         400 
                         × 
                         
                           10 
                           3 
                         
                       
                       N 
                     
                      
                     
                         
                     
                      
                     kHz 
                   
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
           
         
       
     
       Channel Rejection Performance Simulation: 
       [0231]    Computer simulation can be employed to compute the rejection expression of Equation (79). 
         [0232]    In some embodiments a 20-30 dB rejection can be sufficient for a double-ADC architecture as discussed herein. The following table shows three example configurations that can achieve 20 dB rejection 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
               
               
                 N 
                 β 
                 Δ (kHz) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 512 
                 0.4 
                 500 
               
               
                 1024 
                 0.3 
                 160 
               
               
                 2048 
                 0.2 
                 30 
               
               
                   
               
             
          
         
       
     
       Equivalent Discrete-Time Operations for Filtering and Reconstruction: 
       [0233]    An embodiment utlilizing equivalent discrete-time operations can be described. 
         [0234]    A windowing function can be applied 
         [0000]        y   1 ( n )= w ( n ) y ( n )  (84)
 
         [0000]    where w(n) is given by Equation (80) with T w  given by Equation (82) and a sampling time t can be replaced by a sampling index 
         [0000]    
       
         
           
             
               
                 
                   
                     n 
                     = 
                     
                       t 
                       
                         T 
                         s 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                    
                   
                     
 
                   
                    
                   
                     T 
                     s 
                   
                   = 
                   
                     
                       1 
                       400 
                     
                      
                     
                         
                     
                      
                     μ 
                      
                     
                         
                     
                      
                     s 
                      
                     
                         
                     
                      
                     is 
                      
                     
                         
                     
                      
                     the 
                      
                     
                         
                     
                      
                     sampling 
                      
                     
                         
                     
                      
                     
                       period 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   85 
                   ) 
                 
               
             
           
         
       
     
         [0235]    A FFT can be performed on the resulting signal 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Y 
                       1 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         N 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             
                               - 
                               N 
                             
                             / 
                             2 
                           
                         
                         
                           
                             N 
                             / 
                             2 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           
                             y 
                             1 
                           
                            
                           
                             ( 
                             n 
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                               j2π 
                             
                              
                             
                               k 
                               N 
                             
                              
                             n 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   86 
                   ) 
                 
               
             
           
         
       
     
         [0236]    A rejection mask Σ lεΛ Π 2C (f−f l ) can be applied. This operation can comprise the steps of: finding subcarriers whose indices are in the rejection mask; setting Y′(k)=Y 1 (k) for those subcarriers; and, nullifing Y′(k) for all other subcarriers. 
         [0237]    An inverse Fourier transform can be applied 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       y 
                       ′ 
                     
                      
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         N 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             
                               - 
                               N 
                             
                             / 
                             2 
                           
                         
                         
                           
                             N 
                             / 
                             2 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           
                             Y 
                             ′ 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                          
                         
                            
                           
                             j2π 
                              
                             
                                 
                             
                              
                             k 
                              
                             
                               n 
                               N 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   87 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Signal samples, i.e. y′(n)s, inside the flat portion of the window w(t), i.e. tε[(1−β)T w ,(1−β)T w ], can be sent to a DAC in order to construct a rejection signal y′ (t). 
         [0238]    In theory, the multiplication of two signals is only equivalent in continuous-time and discrete-time domains if the output signal is band-limited. Since w(t) is essentially time-limited, it is essentially not frequency-limited. However, because in an embodiment w(t) can have a bandwidth that is significantly narrower than the sampling bandwidth, i.e. 400 MHz, w(t) can be usefully approximated as a delta function in frequency domain. Under these conditions the continuous- and discrete-time multiplications can be essentially equivalent. 
         [0239]    A FFT is of finite size can sample the input signal spectrum at only certain frequencies. The rejection performance result derived here for the continuous spectrum can represent an averaged performance. 
         [0240]    The operations just described above can construct a rejection signal for the flat portion of a window. A signal in the nonflat portion of the window can require additional compensation that can introduce additional error. Constructing a rejection signal for a non-flat portion of a window can require additional FFT resources. That is, supporting a streaming operation can require overlapping two FFT windows such that their flat portions can be connected together. 
         [0241]    The graph  2300  of  FIG. 23  shows simulated multi-carrier signal power spectrums at different IP3s (or different Ds). Nonlinearity can cause spectrum “shoulders” in adjacent bands. The decibel (dB) difference between the inband signal power and the shoulder can be roughly 2D, or the system dynamic range P DR . 
         [0242]    The graph  2300  illustrates simulated signal power spectra under varying device nonlinearities in a multi-carrier system with subcarrier spacing 100 kHz, β=0.16, number of guard band subcarriers  8  (and number of valid data subcarriers  52 ). Individual curves  2302   2304   2306   2308  are shown for IP3-related distance D values of (respectively) 15 dB, 25 dB, 35 dB, and ∞. 
         [0243]    In some embodiments with a fixed output power, a higher device IP3 can be required in order to reduce adjacent channel leakage. In some embodiments, an IP3 requirement can be reduced by applying a digital predistortion technique and/or process. 
         [0244]    In the foregoing specification, the embodiments have been described with reference to specific elements thereof. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the embodiments. For example, the reader is to understand that the specific ordering and combination of process actions shown in the process flow diagrams described herein is merely illustrative, and that using different or additional process actions, or a different combination or ordering of process actions can be used to enact the embodiments. For example, specific reference to NTSC and/or ATSC and/or DTV embodiments are provided by way of non-limiting examples. Systems and methods herein described can be applicable to any other known and/or convenient channel-based communication embodiments; these can comprise single and/or multiple carriers per channel. The specification and drawings are, accordingly, to be regarded in an illustrative rather than restrictive sense.