Abstract:
A physical channel estimator for a communication system using pilot symbols and an equalizer uses a model of the system in which the impulse response of the physical channel is considered separately from the impulse responses of the pulse shaping filters in the transmitter and receiver of the communication system. The system is modeled as if the signals were propagated first through both pulse shaping filters and then through the physical channel. To estimate the physical channel impulse response, known pilot symbols are transmitted and then sampled. The pilot symbol samples and the known impulse responses of the pulse shaping filters are then used to estimate the physical channel impulse response. In one embodiment, the physical channel impulse response is considered time-invariant over the estimation period and a sufficient number of pilot symbol samples are taken so that the system is overdetermined. A least squares method is then used to estimate the physical channel impulse response from the pilot symbol samples and the known responses of the pulse shaping filters. Further refinements include conditioning the estimated physical channel impulse response to improve performance in low SNR conditions and estimating a DC offset incurred from demodulating the received signal.

Description:
FIELD OF THE INVENTION 
     The present invention relates to communication systems and, more particularly, to wireless digital communication systems that include an equalizer. 
     BACKGROUND INFORMATION 
     Some digital communication systems use equalization to increase accurate detection of transmitted symbols in the presence of intersymbol interference (ISI). Such systems often use “pulse shaping” so that the resulting pulses have a zero value at the symbol interval (e.g., a Nyquist pulse). Pulse shaping ideally, in the absence of channel distortion, prevents sequences of pulses from interfering with each other when being sampled. For example, the shaping may be configured to achieve Nyquist pulses, which are well known. Channel distortion, for example, due to the receipt of the transmitted signal over multiple paths with different delays, causes ISI even when Nyquist pulses are transmitted. Equalization is required to compensate for this ISI so that the transmitted symbols are accurately detected. Such equalization and pulse shaping systems are well known (see for example, U.S. Pat. Nos. 5,414,734 and 5,513,215 for a discussion of equalization and Proakis, DIGITAL COMMUNICATIONS, third edition, McGraw-Hill, 1995 for a discussion of pulse shaping). FIG. 1 is a simplified diagram illustrative of a system  10  that uses pulse shaping and equalization. 
     System  10  includes a transmitter  12 , a receiver  14  with an equalizer  16 . System  10  is a wireless digital system in which transmitter  12  broadcasts radiofrequency (RF) signals that are modulated to include digital information. In this system, transmitter  12  receives symbols x(t), which transmitter  12  modulates and broadcasts. Each symbol generally represents one or more bits. For example, each symbol of a sixteen-level quadrature amplitude modulation (QAM) scheme represents four bits. 
     Receiver  14  then receives, demodulates, and samples the broadcasted symbols. Although omitted from FIG. 1 for clarity, in system  10  receiver  14  generally receives a transmission through more than one transmission path. For example, the multiple paths may be the result of more than one transmitter being used to transmit the signals and/or the transmitted signal from a single transmitter being reflected from nearby structures. Typically, the transmission paths between receiver  14  and the various other transmitters are not equal in length and may be changing over time (due to the receiver being moved while receiving a symbol), thereby resulting in multipath fading and ISI. Equalizer  16  then compensates for ISI as the ISI changes over time. Receiver  14  then outputs the detected symbols {circumflex over (x)}(t). 
     Equalizer coefficients can be computed from an estimate of the channel response where the channel is modeled as in model  20  in FIG.  2 . Equalization, ISI, and fading are discussed in more detail in the aforementioned U.S. Pat. Nos. 5,414,734 and 5,513,215, which are assigned to the same assignee as the present invention. 
     FIG. 2 is a diagram illustrative of a simplified model  20  of system  10 . In this model, transmitter  12  includes a pulse shaping filter  22 . Transmitter  12  generally includes several other components besides pulse shaping filter  22  that can influence the shape of the transmitted waveform, and are omitted from this diagram for clarity. Such effects can be modeled as part of pulse shaping filter  22 . Also, receiver  14  generally includes other filters and components that are omitted from the diagram, but can be modeled as part of pulse shaping filter  28 . Transmitter  12  receives digital information represented by symbols x(t), applies the pulse shaper, and uses the result to modulate a carrier signal. 
     Model  20  also includes a physical channel  24 , which represents the multiple paths of the fading channel (the additional transmitters are omitted for clarity). In model  20 , physical channel  24  is modeled as a filter with a time-variant impulse response. The transmitted signal that is “filtered” by physical channel  24  is then received by receiver  14 . A summer  26  is included in model  20  to add noise n(t) to the received signal. Receiver  14  includes a pulse shaping filter  28 , which outputs a signal y(t) to equalizer  16 . Pulse shaping filters  22  and  28  are configured so that the combined filtering results in a Nyquist pulse when there is no channel distortion or transmitter and receiver effects. In this conventional model, system  10  generates signal y(t) according to definition (1) below: 
     
       
           y ( t )={[ x ( t )* p   t ( t )* h ( t )]+ n ( t )}* p   r ( t )  (1) 
       
     
     where y(t), x(t), p t (t), h(t), and p r (t), respectively, represent the output signal of pulse shaping filter  28 , the symbol to be transmitted, the impulse response of pulse shaping filter  22 , the impulse response of physical channel  24 , and the impulse response of pulse shaping filter  28  in the time domain. The symbol “*” indicates the convolution operation. 
     Some conventional systems (e.g., see Crozier, S. N., Falconer, D. D., Mahmoud, S. A., “Least Sum Of Squared Errors (LSSE) Channel Estimation”,  IEE Proceedings - F,  Vol. 138, No. 4, pp. 371-278, August 1991) estimate the overall channel response (i.e., the response due to the pulse shaping filters as well as the physical channel), with symbols x(t) being input into the system. The overall channel is typically modeled as a finite impulse response (FIR) filter, with a predetermined number of coefficients. The number of coefficients is selected to be sufficient to model the channel response without introducing estimation error that significantly affects the performance of the system. In this type of conventional system, the overall channel is modeled according to definition (2) below: 
     
       
           G ( t,z )=P t ( z ) H ( t,z ) P   r ( z )  (2) 
       
     
     where G(t,z), P t (z), H(t,z), and P r (z), respectively, represent the transfer functions of the overall channel response, the pulse shaping filter  22 , the physical channel  24  and the pulse shaping filter  28  in the z domain. It will be appreciated by those skilled in the art that the transfer function of physical channel  24  is time variant and, hence, is denoted as a function of both t and z in definition  2 . Thus, the overall channel response is also a function of t and z. 
     To estimate the time-varying coefficients of the FIR filter implementing G(t,z), a sequence of known pilot symbols is transmitted periodically. Because of the periodic insertion of the sequence of pilot symbols into the stream of data symbols, the transmitted signal has a frame structure. Each frame consists of a sequence of pilot symbols, followed by the data symbols until the start of the next pilot sequence. 
     To estimate the coefficients of the FIR filter implementing G(t,z) at each frame, the received signal corresponding to the pilot sequence is extracted. The error between the output signal predicted by the model and the observed output signal of the actual system is minimized using iterative or least squares minimization methods to adjust the coefficients of the overall channel FIR filter. For example, the aforementioned paper by Crozier et al. uses a least squares estimation method to determine the coefficients of the overall channel FIR filter. 
     The number of coefficients used in the overall channel FIR filter model is related to the number of pilot symbols required in the estimation. That is, for a given number of coefficients for the overall channel FIR filter model, there is a minimum required number of pilot symbols in the sequence. Generally, the number of pilot symbols in the sequence must be greater than or equal to the number of FIR filter coefficients. Longer pilot symbol sequences decrease the number of data symbols in a frame, thereby decreasing data throughput. 
     Generally, for time-invariant systems, the accuracy of the estimation increases as the number of pilot symbols used in the estimation increases. However, in a time-varying system such as system  10  (FIG.  1 ), the accuracy of the estimation tends to decrease as the number of pilot symbols increases because the increased number of pilot symbols occupies a greater timespan, thereby allowing more time for the channel characteristics to change while being estimated. Thus, in selecting the number of coefficients for the overall channel FIR filter, the designer in effect trades error due to estimation for error due to channel variation. Also, because the estimation is typically performed in software by a processor, the computational load on the processor increases as the number of coefficients increases. Accordingly, there is a need for an equalization system that achieves relatively high accuracy with a reduced number of estimated channel coefficients. 
     SUMMARY 
     In accordance with the present invention, a physical channel estimator is provided for a communication system using pilot symbols and an equalizer. In one aspect of the invention, the impulse response of the physical channel is considered separately from the impulse responses of the pulse shaping filters in the transmitter and receiver of the communication system. The system is modeled as if the signals were propagated first through both pulse shaping filters and then through the physical channel. 
     Because the timespan of the physical channel impulse response is typically much less than the timespan of the pulse shaping filter impulse responses, the physical channel can be accurately approximated with an FIR filter having a relatively small number of coefficients (compared to conventional systems that model the overall channel response). This relatively small number of coefficients allows a relatively small number of pilot symbols to be used in estimating the physical channel impulse response, thereby advantageously reducing the time that the physical channel has to vary during the estimation period and freeing more bandwidth for the transmission of data symbols. 
     To estimate the physical channel impulse response, known pilot symbols are transmitted and the corresponding received signal sampled. The pilot symbol samples and the known impulse responses of the pulse shaping filters are then used to estimate the physical channel impulse response. The physical channel impulse response is considered time-invariant over the estimation period. A sufficient number of pilot symbols are used so that the system is overdetermined and a least squares method is then used to estimate the physical channel impulse response from the pilot symbol samples and the known responses of the pulse shaping filters. The relatively small number of physical channel FIR filter coefficients and pilot symbol samples also advantageously reduces the burden on the processing system implementing the channel estimator. 
     In another aspect of the present invention, a cost function scheme is used to condition the estimation in the presence of noise in the received signal. In yet another aspect of the present invention, the effect of an inaccuracy (i.e., DC offset) of an analog demodulator in the receiver is incorporated into the estimation of the physical channel impulse response. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated by reference to the following detailed description, when taken in conjunction with the accompanying drawings. 
     FIG. 1 is a diagram illustrative of a wireless communication system using equalization. 
     FIG. 2 is a diagram illustrative of a conventional model of the system depicted in FIG.  1 . 
     FIG. 3 is a diagram illustrative of a model of the system depicted in FIG. 1, according to one embodiment of the present invention. 
     FIG. 4 is a diagram illustrative of a frame, according to one embodiment of the present invention. 
     FIG. 5 is a block diagram illustrative of an equalizer employing a physical channel estimator, a physical channel interpolator, and a decision feedback equalizer, according to one embodiment of the present invention. 
     FIG. 6 is a flow diagram illustrative of the operation of a channel estimator, according to one embodiment of the present invention. 
     FIG. 7 is a block diagram illustrative of a DSP system used to implement a channel estimator, according to one embodiment of the present invention. 
     FIG. 8 is a block diagram illustrative of a model of an analog quadrature demodulator with DC offsets. 
     FIG. 9 is a block diagram illustrative of the equalizer of FIG. 5 with analog demodulator DC offset removal, according to one embodiment of the present invention. 
     FIG. 10 is a block diagram illustrative of the equalizer of FIG. 5 with analog demodulator DC offset removal, according to another embodiment of the present invention. 
    
    
     DETAILED DESCRIPTION 
     FIG. 3 is a diagram illustrative of a model  30  of the system depicted in FIG. 1, according to one embodiment of the present invention. Model  30  is essentially the same as model  20  (FIG.  2 ), except that the noise n′(t) is pulse shaped and the position of pulse shaping filter  28  is changed from being positioned after physical channel  24  to being positioned between physical channel  24  and pulse shaping filter  22 . 
     Model  30  was developed as follows. Because the transfer functions of the pulse shaping filters are known and time invariant, only the transfer function of physical channel  24  is unknown. Thus, in model  30 , only the response of physical channel  24  is estimated. The overall channel response is then determined by the convolution of the estimated physical channel response with the pulse shaping filter responses. In particular, the physical channel response and the receiver pulse shaping filter response are assumed to be commutative so that overall physical channel response is modeled according to definition (3) below: 
     
       
           G ( t,z )= P   t ( z ) P   r ( z ) H ( t,z )  (3) 
       
     
     where G(t,z), P t (z), P r (z), and H(t,z) are described in conjunction with definition (2) above. Definition (3) in effect assumes that the impulse response of physical channel  24  does not change significantly while the pilot sequence is being received. 
     Using model  30 , a methodology for estimating the impulse response of physical channel  24  is developed as follows. As shown in FIG. 3, pulse shaping filter  28  outputs a signal u(t). Signal u(t) can be determined according to definition (4) below: 
       U ( z )= P   t ( z ) P   r ( z ) X ( z )  (4) 
     where U(z) and X(z) are the z transforms of signals x(t) and u(t). Signal u(t) is propagated through physical channel  24  and, thus, the overall channel output signals can be determined according to definition (5) below: 
     
       
           y ( t )= u ( t )* h ( t )+ n ( t )  (5) 
       
     
     where h(t) represents the impulse response of physical channel  24  and n(t) represents additive receiver noise. 
     In one embodiment of the invention, an FIR filter is used to model the impulse response of physical channel  24 . Consequently, using a physical channel FIR filter of 2j+1 coefficients, signal y(t) outputted by physical channel  24  can be approximated according to definition (6) below:                y        (   t   )       =         ∑     m   =     -   j       j                       h     t   ,     m        T   2                u        (     t   -     m        T   2         )           +     n        (   t   )                 (   6   )                                
     where h t,mT/2  represents one of the samples of the estimated physical channel impulse response at time t. The h t,mT/2  samples have T/2 spacing, where T represents the period of time between symbols. In one embodiment, 2j+1 is set to five (i.e., j=2), so that the impulse response of physical channel  24  is approximated by an FIR filter having five coefficients. Thus, the approximated impulse response spans a duration of about 5T/2 or about a maximum of two and one-half symbols. The relatively short timespan of the approximated physical channel impulse response advantageously tends to reduce the effect of assuming a constant physical channel during the estimation period. 
     For k samples of the received pilot symbols (in T/2 spaced samples), and assuming time invariance during the estimation period, definition (6) may be written in matrix form according to definition (7) below. In one embodiment, k is equal to twenty.                [           y        (     T   2     )                 y        (     2        T   2       )                 y        (     3        T   2       )               ⋮             y        (     k        T   2       )             ]     =            [           u        (       T   2     -     j        T   2         )             u        (       T   2     -       (     j   -   1     )          T   2         )           ⋯         u        (       T   2     +     j        T   2         )                 u        (       2        T   2       -     j        T   2         )             u        (       2        T   2       -       (     j   -   1     )          T   2         )           ⋯         u        (       2        T   2       +     j        T   2         )                 u        (       3        T   2       -     j        T   2         )             u        (       3        T   2       -       (     j   -   1     )          T   2         )           ⋯         u        (       3        T   2       +     j        T   2         )               ⋮       ⋮       ⋮       ⋮             u        (       k        T   2       -     j        T   2         )             u        (       k        T   2       -       (     j   -   1     )          T   2         )                         u        (       k        T   2       +     j        T   2         )             ]     ·     [           h     j        T   2                   h       (     j   -   1     )          T   2                                 ⋮             h       -   j          T   2               ]                 (   7   )                                
     Definition (7) above can be written as shown in definition (8) below: 
     
       
           Y=UH+N   (8) 
       
     
     where Y, U, H, and N are y(t), u(t), h(t), and n(t) in vector and matrix form. Then assuming physical channel  24  is time-invariant during estimation, the impulse response of physical channel  24  can then be estimated using least squares estimation methods, such as disclosed in S. Haykin, ADAPTIVE FILTER THEORY, third edition, Prentice Hall, 1996. Using such methods, the impulse response of physical channel  24  can be estimated according to definition (9) below: 
     
       
           Ĥ =( U*U ) −1   U*Y   (9) 
       
     
     where Ĥ represents the estimate of h(t) in matrix form and U* represents the conjugate transpose of matrix U. By defining the quantity (U*U) −1 U* as matrix R, the estimated physical channel impulse response can be calculated as the product of precomputed matrix R and the vector of received samples. Matrix R can be precomputed because the impulse responses of pulse shaping filters  22  and  28  are known. Thus, definition (9) may be rewritten as definition (10) below: 
     
       
           Ĥ=RY   (10) 
       
     
     Model  30  and definitions (6)-(10) may be applied in a communication system similar to system  10  (FIG.  1 ). A transmitter such as transmitter  12  (FIG. 1) may be used to broadcast the symbols, preferably using a linear modulation scheme. For example, a suitably configured model T9000 transmitter available from Glenayre Electronics, Inc., Charlotte, N.C. may be used. 
     As is well known, the symbols to be broadcast can be grouped into frames. FIG. 4 is a diagram illustrative of a frame, according to one embodiment of the present invention. In this embodiment, frames  40   1 ,  40   2 , and so on, respectively, include pilot symbol sections  41   1 ,  41   2 , and so on. Frames  40   1 ,  40   2 , and so on also, respectively, include data symbol sections  43   1 ,  43   2 , and so on. In a preferred embodiment, each frame includes twelve pilot symbols and thirty-eight data symbols. The twelve pilot symbols for each frame are each sampled twice, providing twenty-four pilot symbol samples (i.e., the y(t) samples in definition (8) at the receiver. Of these twenty-four samples, the middle twenty received samples are used in Y in definition (10) to estimate the physical channel impulse response. Only the middle twenty samples are used to reduce the effects of “data leakage” from the data symbol sections on either side of the pilot symbol section. 
     In a preferred embodiment, the pattern of pilot symbols is essentially a full-spectrum signal within the frequency band of the channel being used for the transmission. For example, in a paging application, a channel may be a frequency band of about 25 kHz. 
     A receiver then receives and processes (e.g., samples, pulse shapes, etc.) the broadcasted symbols in a manner substantially similar to a conventional system. However, in accordance with the present invention, the channel estimation process, which is part of the equalization process, is based on model  30  (FIG. 3) and definitions (6)-(10). FIG. 5 is a block diagram illustrative of an equalizer  50 , according to one embodiment of the present invention. Equalizer  50  includes a physical channel estimator  53 , a physical channel interpolator  55 , and a decision feedback equalizer circuit (DFE)  57 . 
     In one embodiment, DFE  57  is conventional and, thus, is not further discussed herein. For example, the aforementioned U.S. Pat. No. 5,513,215 discloses a DFE. Physical channel interpolator  55  is preferably implemented, as disclosed in co-pending U.S. Pat. No. 6,173,011 C. Rey and O. Katić entitled “Forward-Backward Channel Interpolator”, filed May 28, 1998, and assigned to the same assignee as the present invention. However, in alternative embodiments, any suitable conventional interpolator with linear phase response may be used to implement physical channel interpolator  55 . 
     In this embodiment, physical channel estimator  53  is implemented with a model 1620 DSP processor available from Lucent Technologies. In a preferred embodiment, the DSP processor has on-chip nonvolatile memory to store software programming to estimate the physical channel impulse response according to the methodology described above in conjunction with FIGS. 3 and 4. FIG. 6 is a flow diagram illustrative of the general operation of physical channel estimator  53 , according to one embodiment of the present invention. In this embodiment, a step  61  is first performed in which matrix U, as in definition (7) is determined from the known responses of pulse shaping filters  22  and  28  (FIG.  3 ). Because this embodiment uses twenty pilot symbol samples and five physical channel FIR filter coefficients, matrix U has twenty rows and five columns. 
     In a next step  62 , matrix R is computed from matrix U according to definitions (9) and (10) above. Matrix R is then stored in a memory that is accessible by the DSP. Once pulse shaping filters  22  and  28  (FIG. 3) are set, matrix R is computed only once and is used in estimating the physical channel impulse response on a frame-by-frame basis. In this embodiment, steps  61  and  62  are precomputed. 
     In a next step  63 , the received signal samples of at least ten frames are stored. A step  64  starts a loop through all of the frames in the buffer. The loop is performed as follows. For each frame, in a step  65 , the twenty middle samples of the pilot symbols are extracted. Then in a step  66 , the estimated physical channel impulse response is determined by multiplying matrix R with the vector of twenty pilot symbol samples, according to definition (10). The estimated physical channel impulse response is then stored in a step  67  for use by physical channel interpolator  55  (FIG.  5 ). In a step  68 , the loop counter is incremented and the pointer to the buffer is advanced to point to the next frame. After the completion of the loop, the process returns to step  63  to buffer ten more frames. In one embodiment, the buffering process of step  63  for the next “block” of frames is performed while the current “block” of frames is being processed according to steps  64 - 68 . 
     FIG. 7 is a block diagram of a receiver  70  using physical channel estimator  53  according to one embodiment of the present invention. As described above, a DSP  71  executes a software or firmware program to implement physical channel estimator  53 . DSP  71  also implements other functional blocks of equalizer  50  (FIG.  5 ), such as physical channel interpolator  55  and DFE  57 . A random-access memory (RAM)  73  is used to store data used in estimating the physical channel impulse response. In this embodiment, a 14 kb DRAM device is used to store data in a memory, such as the received pilot symbol samples, and the estimated coefficients of the physical channel FIR filter. A nonvolatile memory (NVM)  75  (e.g., a read-only memory or ROM device) is used to store the precomputed matrix R from definitions (9) and (10). The nonvolatile memory may be part of the on-chip ROM of DSP  71  or, alternatively, a separate memory device. DSP  71  has access to the signal samples generated by the receiver front end (RCVR FE)  79  through an interface unit (IU)  77 . 
     In an alternative embodiment, a cost function scheme is used to reduce the effect of noise on the physical channel estimate in low signal-to-noise ratio (SNR) environments. Cost function techniques are typically used in regression problems when the matrix is poorly conditioned (e.g., see Hager, APPLIED NUMERICAL LINEAR ALGEBRA, Prentice Hall, 1988). In this embodiment, the cost function of definition (11) below is used: 
     
       
           J =( e *) e+ λ( H *) H   (11) 
       
     
     where J represents the cost, e represents the error between the estimated output signal ŷ(t) and the observed output signal y(t), λ is a scalar that represents the weighting of the energy term in the cost function relative to the squared error term, and * represents the conjugate transpose operator. The estimated output signal ŷ(t) is determined from the estimated channel response and matrix U (i.e., by multiplying matrix U by Ĥ). Using least squares techniques to minimize the cost function, the physical channel impulse response may be estimated using definition (12) below: 
     
       
           Ĥ =( U*U+λI   n ) −1   U*Y   (12) 
       
     
     where I n  represents the identity matrix. Comparing definition (12) with definition (9), it can be seen that this cost function technique adds a diagonal of X to the pilot signal correlation matrix U*U before the pilot signal correlation matrix is inverted. 
     It can be shown that this cost function scheme reduces the variance of the estimate of the physical channel impulse response estimate when λ is greater than zero, at the expense of introducing a bias in the estimate. This lower estimation variance tends to improve equalizer performance when the signal is received with a low SNR. However, the bias in the estimate tends to degrade equalizer performance when the signal is received with a high SNR. 
     In one embodiment, the value of λ is predetermined to achieve a desired maximum error floor tolerance at the highest expected SNR, thereby improving the performance of the equalizer over the range of SNR of interest. For example, when the maximum expected          E   b       N   o                            
     (energy per bit over noise spectral density) is 30 dB, λ may be set to about 0.4. This scheme is advantageously used with systems utilizing error correction coding (ECC) because relatively few errors are expected at high SNR and infrequent errors tend to be more easily correctable. Thus, this cost function scheme advantageously provides higher estimation accuracy at low SNR to achieve improved performance during conditions when it is most needed. 
     To implement this embodiment of the cost function scheme into physical channel estimator  53  (FIG.  5 ), definition 10 above (i.e., Ĥ=RY) is used except that matrix R is precomputed according to definition (13) below: 
     
       
           R =( U*U+λI   n ) −1   U*   (13) 
       
     
     In light of the present disclosure, those skilled in the art can implement embodiments that would select among different precomputed R matrices, each computed with a different value of λ according to a measure of the SNR in the received signal. 
     In another aspect of the present invention, the model of the communication system can be modified to include impairments from other sources. For example, some communication systems use quadrature modulation to increase the throughput of the system. Thus, any inaccuracy in the demodulator may affect the accuracy of the physical channel impulse response estimate. When an analog quadrature demodulator (AQDM) is used in the receiver, a DC offset may be introduced in the in-phase (I) and quadrature (Q) output signals of the AQDM. 
     FIG. 8 is a block diagram of a model  80  of an AQDM, that includes the DC offsets. Model  80  includes mixers  81   I  and  81   Q , a phase splitter  83 , a local oscillator  85 , low-pass filters (LPFs)  87   I , and  87   Q , and summers  89   I  and  89   Q . A received signal r(t) is provided to mixers  81   I  and  81   Q . Mixers  81   I  and  81   Q , respectively, then mix the received signal r(t) with signals 2cos(ω c t) and −2sin((ω c t) to recover the I and Q components of received signal r(t). The output signals of mixers  81   I  and  81   Q  are then filtered by LPFs  87   I  and  87   Q  to recover the baseband I and Q component signals. Summers  89   I  and  89   Q , respectively, then add DC offsets I dc  and Q dc  to the output signals of LPFs  87   I  and  87   Q , respectively, generate AQDM output signals I imb  (t) and Q imb  (t). 
     Referring to FIG. 8, the output signal of the AQDM may be modeled according to definition (14) below: 
     
       
           Y   imb   =y   t   +y   dc   (14) 
       
     
     where Y imb  represents the received complex baseband signal at time t, y t  represents the ideal received complex baseband signal at time t, and Y dc  represents the complex DC offset. As is well known, the received complex baseband signal can be expressed in complex notation as in definition (15) below: 
     
       
           y   t   =I   t   +jQ   t   (15) 
       
     
     where I t  represents the I component of y t , j represents the imaginary number {square root over (−1)}, and Q t  represents the Q component of y t . Similarly, the complex DC offset can be represented according to definition (16) below: 
     
       
           y   dc   =I   dc   +jQ   dc   (16) 
       
     
     where y dc  represents the complex DC offset signal at time t, I dc  represents the I component of y dc , and Q dc  represents the Q component of y dc . 
     The above-described embodiments of physical channel estimator  53  do not account for demodulation inaccuracies in estimating the physical channel impulse response. Thus, using definition (9) may cause inaccuracies in the physical channel estimate because the pilot symbol samples will include distortions from the DC offset. 
     In a preferred embodiment, least squares techniques are again applied to estimate the DC offset. By applying definitions (8) and (14), the received complex baseband signal may be modeled according to definition (17) below: 
     
       
         Y imb   =UH+Y   dc   +N   (17) 
       
     
     where Y imb  and Y dc  are vectors that represent the signal y imb  and the DC offset y dc , respectively. More specifically, Y imb  represents [y imb (T/2) y imb (2T/2) . . . y imb (kT/2)] for k samples of received signal symbols. To use the least squares estimation techniques, it is assumed that the DC offset remains constant during the estimation period so that the DC offset may be modeled according to definition (18) below: 
     
       
           Y   dc   =C·o   (18) 
       
     
     where C is a complex constant representing the DC offsets of the I and Q components, and o is a vector of all ones. By substituting the right-hand side of definition (18) into definition (17), definition (19) below may be used to model the physical channel and AQDM DC offset. 
     
       
           Y   imb   =UH+C·o+N   (19) 
       
     
     In one embodiment, the DC offset is estimated and then removed from the complex baseband signal y imb  before estimating the physical channel impulse response. Using least squares techniques, C may be estimated according to definition (20) below: 
     
       
           Ĉ=KY   imb   (20) 
       
     
     where Ĉ is the estimate of C in definition (18) and K is represented by definition (21) below:              K   =         o   T     ·     (       I   n     -     U   ·   R       )                o        2     -       o   T     ·   U   ·   R   ·   o                 (   21   )                                
     where T represents the transpose operation and ∥o∥ 2  is represented by definition (22) below: 
     
       
           ∥o∥=o   T   o   (22) 
       
     
     It will be appreciated that matrix K can be precomputed and stored. Then for a block of received signal frames, the estimated DC offset Ĉ can be determined for each pilot sequence in the captured block of frames using definition (20). 
     In one embodiment, a vector CC is created using m estimates Ĉ i  (where i=1,2, . . . , m) of the DC offsets that were determined from m frames of data. In the preferred implementation m may be set to ten. Then the average value of vector CC represents a constant DC offset for the whole received block by definition (23) below:                  C   ^     ave     =       1     m                         ∑     i   =   1     m                       C   ^     i                 (   23   )                                
     where Ĉ ave  represents the average value of vector CC. Ĉ ave  is subtracted from a vector Y imb  (representing a vector of the received signal samples) as represented by definition (24) below: 
     
       
           Y=Y   imb   −Ĉ   ave   (24) 
       
     
     where Y represents a vector of the received signal samples without DC offset. Then the baseband received signal Y is used in channel estimator  53 . In this manner, the DC offset is advantageously removed from the signal before being processed by the estimator to increase accuracy. 
     FIG. 9 is a block diagram illustrative of an equalizer  90  incorporating the analog demodulator DC offset scheme described above. Equalizer  90  can be used as an alternative to equalizer  50  (FIG.  5 ). This embodiment of equalizer  90  is substantially similar to equalizer  50  (FIG.  5 ), but with the addition of a subtractor  92  and a DC offset estimator  94 . More specifically, subtractor  92  and DC offset estimator  94  are connected to receive the received signal samples y imb . Subtractor  92  is also connected to receive the output sample generated by DC offset estimator  94 . CE  53 , CI  55 , and DFE  57  are connected as in equalizer  50  (FIG.  5 ), except that they operate on the output sample generated by subtractor  92  instead of received signal samples y imb . 
     Equalizer  90  operates as follows. DC offset estimator  94  is configured to determine Ĉ ave  according to definition (23) above. Subtractor  92  then subtracts Ĉ ave  from the received signal vector Y imb  to generate vector Y, according to definition (24). This vector Y is then received by physical channel estimator  53 , which then generates the physical channel impulse response estimates as previously described. As a result, accuracy is improved because the DC offset was subtracted from the received signal samples before being used to generate the physical channel impulse response estimates and before being input into DFE  57 . 
     In another embodiment, the estimation of the DC offset and the estimation of the physical channel impulse response are performed simultaneously. Using least squares, H may be estimated according to definition (25) below: 
     
       
           Ĥ=R   2 Y imb   (25) 
       
     
     where R 2  is represented according to definition (26) as:                R   2     =     R   -       R   ·   o   ·     o   T     ·     (       I   n     -     U   ·   R       )                o        2     -       o   T     ·   U   ·   R   ·   o                   (   26   )                                
     where o is the aforementioned matrix of ones. Matrix R 2  can also be precomputed and stored in memory. Then physical channel estimator  53  would be configured to generate the physical channel impulse response estimates according to definition (25). 
     FIG. 10 is a block diagram illustrative of an equalizer  100  with analog demodulator DC offset removal, according to another embodiment of the present invention. Equalizer  100  is substantially similar to equalizer  90  (FIG.  9 ), except that CE  53  is replaced with a CE  102 . In addition, CE  102  is connected to receive received signal samples y imb  instead of the output samples of subtractor  92  as in equalizer  90 . Equalizer  100  determines the DC offset and the estimated channel response simultaneously, in accordance with definitions (20)-(21) and (25)-(26). More specifically, DC offset estimator  94  determines the DC offset according to definitions (20) and (21), while CE  102  concurrently determines the estimated channel response directly from received signal samples y imb  according to definitions (25) and (26). Then, the estimated DC offset may be averaged using definition (23) as described above for equalizer  90  and subtracted from the received signal vector Y imb  in subtractor  92  to generate vector Y, according to definition (26). 
     The embodiments of the channel estimator described above are illustrative of the principles of the present invention and are not intended to limit the invention to the particular embodiments described. For example, in light of the present disclosure, those skilled in the art can devise other implementations using different DSPs or general-purpose processors. Other embodiments of the present invention can be adapted for use in communication systems other than the described wireless mobile communication applications. Accordingly, while the preferred embodiment of the invention has been illustrated and described, it will be appreciated that various changes can be made therein without departing from the spirit and scope of the invention.