Patent Publication Number: US-8971398-B2

Title: System and method for generating a radio frequency pulse-width modulated signal

Description:
TECHNICAL FIELD 
     This invention relates generally to semiconductor circuits and methods, and more particularly to a system and method for generating a radio frequency pulse-width modulated signal. 
     BACKGROUND 
     Pulse-width modulated systems are used to provide high efficiency amplification and transmission in applications that vary widely from low-power consumer audio devices, such as MP3 players, to high power data transmission circuits such as base station transmitters. High efficiency is achieved by minimizing power losses due to bias current within the output stages of an amplifier. For example, in audio applications, a class-D amplifier is used to switch the terminals of a loudspeaker between two supply voltages at a frequency greater than the bandwidth of the desired output signal. Here, high frequency switching energy is filtered by the characteristics of the load circuit, for example, the inductance of the loudspeaker. Similarly, in RF applications, a power amplifier (PA), is driven by a pulse-width modulated signal with a pulse frequency greater than the bandwidth of interest. Out-of-band energy is then filtered using an RF bandpass filter, such as a SAW filter for low power applications or a cavity filter for high power applications. Because there is a minimal IR drop across the output stage of devices operating in a switched manner, as can be the case when PWM signals are used, dissipated power across the output stages of the devices are minimized and efficiency is improved. 
     The generation of high dynamic range PWM signals, however, poses a number of challenges at high frequencies and at radio frequencies (RF). As the bandwidths of broadband systems increase, there is a corresponding increase in the switching rate of the PWM signal used to drive the upconverter. Parasitic losses at these high switching rates, however, may lower the overall efficiency of the PWM system. 
     One way to reduce the switching rate in a PWM system is to use multiple parallel PWM signal levels to drive parallel RF power amplifiers. While the overall switching rate may be reduced, other inefficiencies arise such as power losses due to the loading of multiple stages, and losses due to reflected power at the bandpass filter. 
     SUMMARY OF THE INVENTION 
     In an embodiment, a method of producing a multi-level RF signal includes producing plurality of pulse-width modulated signals based on an input signal. The method further includes driving a corresponding plurality of parallel amplifiers with the plurality of pulse-width modulated signals by setting a parallel amplifier to have a first output impedance when a corresponding pulse-width modulated signal is in an active state and setting the parallel amplifier to have a second output impedance when the corresponding pulse-width is in an inactive state. The method also includes phase shifting the outputs of the plurality of parallel amplifiers, wherein phase shifting transforms the second output impedance into a third impedance that is higher than the second output impedance, and combining the phase shifted outputs. 
     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  illustrates a conventional multi-level PWM generator; 
         FIG. 2  illustrates an embodiment multi-level RF transmitter; 
         FIG. 3  illustrates a graph of coding efficiency for embodiment PWM generators; 
         FIG. 4  illustrates an embodiment modulated PWM output spectrum; 
         FIG. 5  illustrates an embodiment multi-level PWM generator using phase shifted reference functions; 
         FIGS. 6   a - c  illustrates waveforms of embodiment PWM generators; 
         FIGS. 7   a - b  illustrate a multi-level RF transmission systems according to further embodiments; 
         FIG. 8  illustrates a waveform diagram of an embodiment multi-level RF transmission system; 
         FIG. 9  illustrates a graph showing output power and efficiency with respect to duty cycle for an embodiment multi-level RF transmission system; 
         FIG. 10  illustrates an embodiment PWM modulator; and 
         FIG. 11  illustrates an embodiment PWM modulator implementation. 
     
    
    
     DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS 
     The making and using of the presently preferred embodiments are discussed in detail below. It should be appreciated, however, that the present invention provides many applicable inventive concepts that can be embodied in a wide variety of specific contexts. The specific embodiments discussed are merely illustrative of specific ways to make and use the invention, and do not limit the scope of the invention. 
     The present invention will be described with respect to preferred embodiments in a specific context, namely a pulse-width modulator and output stage for use with baseband and RF transmission systems. The invention may also be applied, however, to other types of circuits and systems, such as audio systems, communication systems, and other electronic or optical systems that utilize pulse-width modulated signals. 
     In an embodiment, a discrete-time pulse-width modulated signal is created by calculating a finite number of basis functions representing an output PWM signal. The band limited nature of these basis functions prevent aliasing due to the discrete time nature of the signal, thereby allowing for a high-dynamic range and a low out-of-band noise floor. The use of basis functions to generate pulse-width modulated signals to efficiently drive an amplifier (PA) is described in U.S. patent application Ser. No. 13/081,628 filed on Apr. 7, 2011 entitled “System and Method for Generating a Pulse-width Modulated Signal,” which application is incorporated by reference herein in its entirety. 
     In some embodiments, the use of basis functions to create a pulse-width modulated signal is extended to create a multi-level RF signal using multiple basis function derived pulse-width modulated signals. These multiple pulse-width modulated signals are each used to drive an array of power amplifiers whose outputs are combined to form a single output. At this output, each of the pulse-width modulated signals combine to form a multi-level output. In some embodiments, these pulse-width modulated signals are upconverted before driving the power amplifiers. Reflected power is avoided by phase shifting each output of each power amplifier before the outputs are combined, and passing the phase shifted combined output though a circulator or other isolation device. 
     One of the main issues with using a pulse width modulated signal in an RF system is the large spectral out-of-band components located near the desired signal band. In many cases, these close in out-of-band components are filtered out prior to the signal being fed to an antenna, cable or other device in order to meet the out-of-band spectral requirements of the particular system. Filtering these out-of-band components, however, may significantly reduce the efficiency of the PA if the reflected out-off band components from the filter are dissipated in an isolator or other device. 
     One way in which efficiency may be maintained is if the coding efficiency, defined as the ratio of in band power to total power, is increased. One way in which the coding efficiency can be increased is by using multiple switched-mode power amplifiers (SMPAs) in which two or more PAs are driven with individual PWM signals. The PA output ports are connected together with a power combiner prior the signal being fed to an isolator and filter. This concept works well as long as the PWM frequency is kept within the range of a few MHz or lower. In the future, however, high-bandwidth base station applications may require bandwidths of 100 MHz and above. The efficiency of these high bandwidths systems may be lower due to high PWM frequencies needed to maintain high linearity. These high PWM frequencies are generally difficult to handle on the PA side because of the required high bandwidth matching networks and the increased parasitic losses in the transistors itself. 
     Another issue relates to base stations operating in a duplex mode in which the receive band is quite close to the transmit band. To avoid high interference between the transmit and receive signal paths, a sufficiently high PWM frequency is used. In some proposed embodiments, a PWM coding method is used in combination with a power combiner network that reduces the PWM frequency, relaxes the bandwidth requirements on each individual PA, yet maintains a high enough switching rate at the filter input to ensure high future bandwidth and efficiency requirements. 
       FIG. 1  illustrates a conventional system  100  for generating a multi-level PWM signal. Here, the incoming envelope of a complex modulated signal  104  is separated in the amplitude domain by N−1 amplitude and phase shifted triangular reference functions  102  and  106 . Comparators  108  and  110  compare a respective reference function with the input signal  104  and generate PWM respective PWM signals  112  and  114 . While only two reference functions  102  and  106  and two comparators  108  and  110  are illustrated for simplicity of illustration, any number of comparators may be used depending on the particular application and its specifications. 
       FIG. 2  illustrates an embodiment multi-level PWM system  130  adapted for producing a multi-level RF output. In an embodiment, N−1 individual PWM baseband signals generated by comparators  108  and  110  are upconverted by mixers  116  and  118 , respectively, to form upconverted signals  126  and  128 , which are then amplified by PAs  120  and  122 . Two PAs  120  and  122  are shown for simplicity of illustration; however, more PAs may be used to amplify N−1 upconverted PWM signals. The outputs of PAs  120  and  122  are combined by combiner  124  to form N-level multi-level RF output signal  132 . In some embodiments, high efficiency is achieved by driving PAs  120  and  122  in a highly efficient manner, for example, at a high signal level. In an embodiment, PAs  120  and  122  are driven into 1 dB compression to get a high peak efficiency. Alternatively, the PA&#39;s may be driven at other levels. The resulting combined output  132  is an N-Level PWM signal. 
     After the amplification, individual RF-bursts (PA output signals) are combined together by combiner  124  to build N-level PWM signal  132 . It should be appreciated that as the number of PWM levels increases, the out-off band spectral energy of N-level PWM signal  132  decreases. This increases the overall efficiency of the transmitter for complex modulated signals with high peak to average power ratios (PAPR) even if an isolator is used in front of the required bandpass filter. 
     In an embodiment, the total signal (PWM) is amplified efficiently because PA  120  and  122  are either in an off state or completely on. However, PWM signal  132  is filtered using, for example, a bandpass filter before the antenna to reconstruct or demodulate the original modulated signal. In order to reconstruct the original modulated signal, out-of-band spectral components are filtered out. These filtered components are reflected by the bandpass filter input and travel back to an isolator between the combiner and the bandpass filter, where they are dissipated. This dissipation causes a decrease in efficiency. Using a higher coding efficiency reduces out-of-band power, thereby reducing this efficiency degradation. A higher efficiency is, therefore, achieved. 
       FIG. 3  illustrates a graph of the coding efficiency of signals produced by the system illustrated in  FIG. 2 . Curve  150  represents the coding efficiency of a two-level PWM scheme using a single PA, and curve  152  represents a three level PWM scheme using two PA. As can be seen, coding efficiency for a two level PWM  150  decreases linearly with lower input voltage (input signal in  FIG. 2 ), while the coding efficiency for the three level PWM  152  stays high down to the half of the maximum input voltage. This increase in coding efficiency also leads to an improvement in the electrical efficiency for modulated signals with high PAPR. As the number of PWM levels is increased beyond the three levels, as shown in  FIGS. 1 and 2 , the coding efficiency and, hence, the electrical efficiency can be further improved by using more hardware. 
     The architecture in  FIG. 2  works effectively for lower signal bandwidths, for example, below 20 MHz. At higher signal bandwidths, such as 75 MHz and above, the required relative bandwidth for the input matching networks of PAs  120  and  122  shown in  FIG. 2  are close to or even higher than one. This required relative bandwidth may be defined, in some embodiments, as:
 
Bs×L×Nh,
 
where Bs is the signal bandwidth, L is a factor relating to required in band signal quality (linearity), and Nh is the number harmonics needed for high efficiency PA operation. In one embodiment, L=6 and Nh=4. These values are estimated numbers that depend on the particular application. Other values may be used for L and Nh depending on the particular system and its specifications.
 
       FIG. 4  depicts a spectral plot for a 3-level PWM signal having a 900 MHz carrier frequency, a 35 MHz signal bandwidth and four harmonics. To obtain 70 dBc input signal dynamic range, a PWM reference frequency of about 150 MHz is needed. This is difficult to implement using state of the art technologies because a high bandwidth, for example (4×2×150) MHz=1200 MHz, for the PAs, would be needed. This high bandwidth is difficult to match at the 1 GHz or 2 GHz carrier frequencies used for radio base station applications because the relative bandwidth is too high. 
       FIG. 5  illustrates multi-level PWM transmitter  200  according to a further embodiment of the present invention. In contrast to the system of  FIGS. 2 and 3 , the multi-level PWM signal here is generated with phase shifted reference functions  202  and  204  that are phase shifted with respect to each other, but have an equal amplitude level. Reference functions  202  and  204  are compared to input signal  206  using comparators  208  and  210  to produce PWM signals  212  and  214 , respectively. It should be appreciated that, while only two reference functions, comparators and PWM signals are shown for simplicity of illustration, any number of reference functions, comparators and PWM signals may be used in other embodiments. 
     It is important to note that the PWM generation is not limited by the system illustrated in  FIG. 5 . The PWM generation can also be performed using basis functions described in U.S. patent application Ser. No. 13/081,628. For example, in an embodiment, a band limited kernel or basis function can be derived natural-sampling trailing-edge PWMs for input signals x(t) bounded between 0 and 1. The continuous time input-output relationship of such a PWM signal can be written as: 
                     y   ⁡     (   t   )       =       ⁢     f   ⁢     {     x   ⁡     (   t   )       }                     =       ⁢       x   ⁡     (   t   )       +       ∑     k   =   1     ∞     ⁢           ⁢       1     k   ⁢           ⁢   π       ⁢     (       sin   ⁡     (     2   ⁢   π   ⁢           ⁢     kf   P     ⁢   t     )       -     sin   ⁡     (       2   ⁢   π   ⁢           ⁢     kf   P     ⁢   t     -     2   ⁢   π   ⁢           ⁢     kx   ⁡     (   t   )           )         )             ,               
where x(t) is the input signal, f p  is the PWM frequency, and y(t) is the pulse-width modulated output signal. From the above equation, it is apparent that the kernel is not band-limited, since y(t) has frequency components approaching infinity when x(t) is non-zero.
 
     The above expression can be expressed in a discrete-time or sampled version where x(nT) is substituted for x(t): 
                     y   ⁡     (   nT   )       =       ⁢     f   ⁢     {     x   ⁡     (   nT   )       }                     =       ⁢       x   ⁡     (   nT   )       +       ∑     k   =   1     ∞     ⁢           ⁢       1     k   ⁢           ⁢   π       ⁢     (       sin   ⁡     (     2   ⁢   π   ⁢           ⁢     kf   P     ⁢   nT     )       -     sin   ⁡     (       2   ⁢   π   ⁢           ⁢     kf   P     ⁢   nT     -     2   ⁢   π   ⁢           ⁢     kx   ⁡     (   nT   )           )         )             ,               
where T is the sampling period. In the above equation, aliasing occurs for T&gt;0. Therefore, in some embodiments, the kernel is band-limited by using only k terms instead of all terms:
 
                     y   ⁡     (   nT   )       =       ⁢       f   BL     ⁢     {     x   ⁡     (   nT   )       }                   =       ⁢       x   ⁡     (   nT   )       +       ∑     k   =   1     K     ⁢           ⁢       1     k   ⁢           ⁢   π       ⁢       (       sin   ⁡     (     2   ⁢   π   ⁢           ⁢     kf   P     ⁢   nT     )       -     sin   ⁡     (       2   ⁢   π   ⁢           ⁢     kf   P     ⁢   nT     -     2   ⁢   π   ⁢           ⁢     kx   ⁡     (   nT   )           )         )     .                       
Therefore, for a sufficiently short sampling period T, the aliasing problem is avoided. In embodiments, the PWM signal is generated based on the band-limited kernel from the last equation, where each harmonic k is calculated according to:
 
     
       
         
           
             
               
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     One difference between the multi-level PWM generation system illustrated in  FIG. 1  and the multi-level PWM generation system illustrated in  FIG. 5 , is that the PWM generation system of  FIG. 1  produces a thermometer coded output, while the system of  FIG. 5  does not. In a system having a thermometer coded output, comparators used to detect higher signal levels are only switched on if comparators used to detect lower signal levels are already active. This is not the case in the PWM scheme shown in  FIG. 5 . Here, each comparator may be switched on independent of the detection state of other comparators. By allowing for independent comparator switching, the effective switching rate of the composite output PWM signal is N times the switching rate of the individual comparator and PA. 
       FIGS. 6   a  and  6   b  illustrate waveform diagrams illustrating the time domain performance of thermometer coded multi-level PWM generation and non-thermometer coded multi-level PWM generation.  FIG. 6   a  illustrates waveforms generated by the system of  FIG. 1  that uses amplitude separated reference functions. Curves  302  and  304  represent the comparator outputs, curve  306  represents the combined multi-level PWM signal, and curve  308  represents the filtered PWM output signal.  FIG. 6   b  illustrates waveforms generated by the system of  FIG. 5  that uses phase shifted reference functions. Curves  312  and  314  represent the comparator outputs, curve  316  represents the combined multi-level PWM signal, and curve  318  represents the filtered PWM output signal. 
     It can be seen that, in the amplitude separated reference function system of  FIG. 1 , curve  304  is high only if curve  302  is high, whereas, in the phase separated reference function system of  FIG. 5 , curves  312  and  314  are active independently to each other. Furthermore, it can be seen that multi-level PWM signal  306  (N=3) generated by the amplitude separated reference function system of  FIG. 1  has an effective switching rate of the switching rate used to generate the PWM signal (i.e. 80 MHz). On the other hand, multi-level PWM signal  316  (N=3) generated by the phase shifted reference function system of  FIG. 5  has an effective switching rate of twice the switching rate used to generate the PWM signal (i.e. 160 MHz). In some embodiments, this higher effective switching rate is double the switching rate of the individuals PAs. 
     In an embodiment, this higher effective switching rate also results in a PWM spectrum where the first harmonics are shifted away from the desired signal band at two times the PWM switching frequency without requiring a doubling of the PA matching network bandwidth. This spectral behavior is illustrated in  FIG. 6   c  for a system having a 35 MHz signal bandwidth and 80 MHz PWM reference frequency. Curve  320  represents the output spectrum of a system using amplitude separated reference functions, for example the system of  FIG. 1 . Curve  322  represents the output spectrum of a system using phase shifted reference functions, for example the system of  FIG. 5 . As shown, the amplitude separated reference function system achieves about a 30 dBc in-band dynamic range, while the phase shifted reference function system achieves about a 60 dBc in-band dynamic range. It should be appreciated that this example is just an illustration of one embodiment system comparison. Actual performance may vary according to the particular embodiment and its implementation. 
     While it is possible to improve the dynamic range by using phase shifted basis reference functions to increase the effective output PWM switching rate, RF combiner structures, such as the Doherty combiner, poses some issues with the combining the individual modulated PWM signals to form a final multi-level RF signals. In the particular situation where PWM signal  212  is active, but PWM signal  214  is inactive, the Doherty loses efficiency because PA  122  is switched off and the high impedance of PA  122  is transformed via the 90 degree transmission line of the Doherty combiner to a short on the combiner output. This prevents RF power from flowing from upper PA  120  to the load, which considerably degrades the efficiency of the system. 
       FIG. 7   a  illustrates multi-level RF system  400  according to an embodiment of the present invention. Multi-level RF system  400  has a modulator  404  that generates a plurality of PWM signals  430  and  432  from input baseband signal  402 . In some embodiments, modulator  404  uses phase shifted reference functions of the same amplitude to generate PWM signals  430  and  432 , and operates according to systems and method discussed in described in U.S. patent application Ser. No. 13/081,628. PWM signals  430  and  432  are upconverted by upconverters  406  and  408 , respectively, the outputs of which are amplified by PAs  410  and  412 . The outputs of PAs  410  and  412  are combined by a network having 90 degree phase shifters  418  and  420 , and switches  414  and  416  that couple the output of PAs  410  and  412  to a RF ground. In an embodiment, 90 degree phase shifters  418  and  420  are implemented by quarter-wavelength transmission lines having an impedance of √{square root over (2)}R L , where R L  is the output impedance of the PAs. In the embodiment of  FIG. 7   a , R L  is about 50 Ohms. Alternatively, other impedances can be used. The outputs of phase shifters  418  and  420  are coupled to an input port of circulator  422 , which is terminated by termination  424 . Bandpass filter  426  is coupled between the output port of circulator  422  and antenna  428 . Alternatively, an isolator or other device that decouples PAs  410  and  412  from bandpass filter  426  may be used. 
     In system  400 , PAs  410  and  412  may be in one of the four states 00,01,10,11. If both PAs are off (00), no output power is delivered to the antenna and no DC power is needed for this operation, thereby providing for efficient operation. If both PAs are on (11) or in an active state, the combiner operates as a Wilkinson combiner that combines the output power of the two PAs without losses (ideal case). If only one of the PA is on and the other is off (01,10), the “off” path is decoupled from the “on” path. This can be accomplished with the switches  414  and  416  using a switching rate equal to the PWM reference frequency. For example, if PA  410  is in the “off” state, and its output current is zero (infinite impedance), and switch  414  is coupled between the input of phase shifter  418  to an RF ground. Phase shifter/transmission line  418  transforms the RF ground to an open circuit at the output of phase shifter/transmission line  418 . Likewise, when PA  412  is in an “off” state, switch  416  is coupled between the output of PA  412  to an RF ground, which is transformed to an open circuit by phase shifter/transmission line  420 . The decoupling effect of the transformed open circuits helps maintain efficiency operation for the states 01 and 10. The system shown in  FIG. 7  is not limited to creating a three level signal, and may be extended to N levels as described in embodiments above. Switches  414  and  416  may be implemented, for example, using discrete devices such as Shottky diodes, LDMOS transistors, GaN transistors, using integrated solutions in the transistor package of the PA, or even on chip. 
     In an embodiment, system  400  may be extended to have an arbitrary number of baseband PWM signals, upconverters, PAs, switches and phase shifters depending on how many RF output amplitude levels are desired.  FIG. 7   b  illustrates embodiment multi-level RFs system  450  that is configured to transmit N output levels. Here, modulator  454  outputs N−1 output channels  456  to  458  to N−1 upconverters  460  to  462  to produce upconverted signals s 1 (t) to s N-1 (t). N−1 PAs  464  to  466  amplify upconverted signals s 1 (t) to s N-1 (t), the outputs of which are combined by N−1 90 degree phase shifters  468  to  470 . N−1 switches  472  to  474  selectively couple the outputs of PAs  464  to  466  to RF ground. In an embodiment, 90 degree phase shifters  468  to  470  have an impedance of about √{square root over (N−1)}·R L , and may be implemented using quarter-wavelength transmission lines. Operation of phase shifters  468  to  470  with respect to switches  472  to  474  proceeds as described with respect to the embodiment of  FIG. 7   a . In an embodiment, amplifiers that are not transmitting have their outputs coupled to ground via their corresponding switch. For example, if PA  466  is not transmitting, switch  474  is closed. The resulting RF ground is then transformed to an RF open at the output of 90 degree phase shifter  470 . 
       FIG. 8  illustrates waveform diagrams associated with system  400  of  FIG. 7 . Curve  502  represents the input baseband signal, curves  504  and  506  represent phase shifted reference signals used in the modulator  404 , curve  508  represents the combined multi-level RF output. Curves  510  and  512  represent the input of PAs  410  and  412 , respectively, and curves  518  and  520  illustrate the output of PAs  410  and  412 , respectively. The switch control signals for PAs  410  and  412  are  514  and  516 , respectively. In an embodiment, the system has a constant input signal of 0.7 times the maximum duty cycle (Umax) of baseband input signal ( FIG. 7 ). Umax may also be defined, for example, as the normalized maximum input magnitude to the modulator  404  ( FIG. 7 ), where the maximum input magnitude is normalized to 1. If only one of PAs  410  and  412  is active, the corresponding driving signal ( 510  or  512 ) is halved to avoid a strong compression within the PA, because the PAs sees twice the optimum load impedance in these conditions. In an embodiment, if only one PA is active, the modulator generates half of the maximum PWM signal magnitude compared to the case if both PAs are active. 
       FIG. 9  illustrates a graph of overall efficiency  550  and output power  552  of the transmitter shown in  FIG. 8  for an input signal whose amplitude is swept from zero to Umax (duty-cycle). 
       FIG. 10  illustrates an embodiment band limited function generator  630  for one basis function where k=5, meaning that the PWM signal is generated using the first 5 Fourier harmonics. Blocks  632 ,  634 ,  636 ,  638  and  640  generate functions g 1 { }, g 2 { }, g 3 { }, g 4 { } and g 5 { }, respectively. Input signal x(nT) is summed with the outputs of g 1 { }, g 2 { }, g 3 { }, g 4 { } and g 5 { } using summing blocks  642 ,  644 ,  646 ,  648  and  650 . It should be appreciated that in alternative embodiments, greater or fewer harmonics than five can be used. It should be further appreciated that function generator  630  shown in  FIG. 10  is just one example of show band limited functions can be generated. In alternative embodiments, other functionally equivalent structures can be used. In an embodiment, each of the multiple basis functions may be implemented using function generator  630 . 
       FIG. 11  illustrates embodiment architecture  700  used to generate plurality of embodiment band limited functions. Here, memory  706  implements a lookup table in which basis functions g k { } are accessed using digital input signals x 1 (n) to x N (n) as an addresses. Address decoders  702  to  704  produce decoded address lines of width N LUT  to memory  706 , where N is the number of PWM levels. Adders  708  and  710  sums outputs of memory  706  to form digital output word y 1 (n) to y N (n). In one embodiment, digital output word y(n) is converted into the analog domain via digital-to-analog converters (DACs)  712  and  714  to form y 1 (t) to y N (t). In alternative embodiments, further processing, for example, a digital upconversion, is performed on digital output words y 1 (n) to y N (n) before performing a digital-to-analog conversion. In some embodiments, DACs  712  and  714  can be omitted. 
     In an embodiment, one address decoder, adder and DAC is included for each basis function. To generate an N-level PWM signal, N address decoders, N address and N DACs are implemented. The size of memory is, therefore, N×2 N     LUR     +1 . The width of the busses for x 1 (n) to x N (n) and y 1 (n) to y N (n) are m bits wide. In one embodiment, m=8, N LUT =5, and N=1. Alternatively, other values may be used depending on the particular embodiment and its specifications. 
     In some embodiments, the band limitation of the PWM generator creates an approximation of an ideal PWM signal, which means that the signal contains not only temporal pulse-width information but also amplitude information. Therefore, a low resolution DAC is used to convert this multi-bit digital signal into the analog domain in some embodiments. Depending on the required signal quality or dynamic range for a particular implementation, a resolution of 8 to 10 bits for a 60 dB to 70 dB dynamic range is sufficient. Alternatively, greater or fewer than 8 or 10 bits can be used depending on the implementation and its specifications. 
     In an embodiment, a method of producing a multi-level RF signal includes producing plurality of pulse-width modulated signals based on an input signal; where each of the plurality of pulse-width modulated signals include an active state and an inactive state. The method also includes driving a corresponding plurality of parallel amplifiers with the plurality of pulse-width modulated signals, where driving includes setting a parallel amplifier to have a first output impedance when a corresponding pulse-width modulated signal is in the active state, and setting the parallel amplifier to have a second output impedance when the corresponding pulse-width is in the inactive state. The method further includes phase shifting the outputs of the plurality of parallel amplifiers combining the phase shifted outputs. In an embodiment, phase shifting transforms the second output impedance into a third impedance, where the third impedance is higher than the second impedance. 
     In an embodiment, combining the phase shifted outputs includes coupling the phase-shifted outputs to an isolation device, such as a circulator. In some embodiments, the isolation device decouples the PAs from a bandpass filter, or other circuit, coupled to an output of the isolation device. In some embodiments, setting the parallel amplifier to have the second output impedance includes coupling an output of the parallel amplifier to an RF short. In an embodiment, the phase shifting transforms the RF short to an RF open. Furthermore, producing the plurality of pulse-width modulated signals may include forming a multi-level pulse-width modulated signal. The method may also include upconverting the plurality of pulse-width modulated signals before driving the corresponding plurality of parallel amplifiers. 
     In an embodiment, producing the plurality of pulse-width modulated signals includes generating a first set of Fourier harmonics approximating a pulse-width modulated signal having a pulse-width proportional to a first digital input signal, and generating a second set of Fourier harmonics approximating a pulse-width modulated signal having a pulse-width proportional to a first digital input signal. The second set of Fourier harmonics is phase shifted from the first set of Fourier harmonics. 
     In an embodiment, the method further includes producing a first pulse-width modulated output signal based on the first set of Fourier harmonics, and producing a second pulse-width modulated output signal based on the second set of Fourier harmonics. Producing the first and second pulse-width modulated signals may include performing digital-to-analog conversions. In some embodiments, producing the plurality of pulse-width modulated signals includes calculating a finite number of first basis functions of a first pulse-width modulated signal based on the input signal, and forming a first electronic output signal based on the calculated finite number of first basis functions, where the first electronic output signal forms a first of the plurality of pulse-width modulated signals. Producing the plurality of pulse-width modulated signals also includes calculating a finite number of second basis functions of a second pulse-width modulated signal based on the input signal and forming a second electronic output signal based on the calculated finite number of second basis functions. The second basis functions are phase shifted from the first basis functions, and the second electronic output signal forms a second of the plurality of pulse-width modulated signals. 
     In an embodiment, a method of generating a multi-level pulse-width modulated signal includes calculating a finite number of first basis functions of a first pulse-width modulated signal based on an input signal, forming a first electronic output signal based on the calculated finite number of first basis functions, calculating a finite number of second basis functions of a second pulse-width modulated signal based on the input signal, and forming a second electronic output signal based on the calculated finite number of second basis functions. The second basis functions are phase shifted from the first basis functions. 
     In an embodiment, the method further includes driving a first output driver with the first electronic output signal, and driving a second output driver with the second electronic output signal. In some embodiments, the method further includes upconverting the first electronic output signal before driving the first output driver, and upconverting the second electronic output signal before driving the second output driver. 
     In an embodiment, driving the first and second output drivers includes configuring the first output driver to have a first impedance when the first electronic signal indicates an active state, configuring the first output driver to have a low impedance when the first electronic signal indicates an inactive state, configuring the second output driver to have a first impedance when the second electronic signal indicates and active state, configuring the second output driver to have a low impedance when the second electronic signal indicates an inactive state, phase shifting outputs of the first and second output drivers, and combining the phase shifted outputs of the first and second output drivers to form a multi-level RF signal. In an embodiment, the low impedance is transformed into a high impedance, and the high impedance is higher than the first impedance. 
     In an embodiment, a transmitter for a multi-level RF signal the transmitter includes a plurality of branches coupled to a common signal port. Each branch contains a driving circuit having an output that has a first output impedance when the driving circuit is active and a second output impedance when the driving circuit is inactive. Each branch also contains a phase shifter coupled between the output of the driving circuit and the common output port, and an isolation device such as a circulator having a first port coupled to the common signal port, a second port coupled to a termination, and an output port. 
     In an embodiment, the second output impedance is an RF short, and the phase shifter transforms the second impedance to a high impedance. In an embodiment, the phase shifter provides a phase shift of about 90 degrees. Furthermore, the phase shifter may have a third impedance. In an embodiment, the plurality of branches include N branches, and the third impedance is about √{square root over (N−1)} times the first impedance. In some embodiments, each driving circuit is configured to be selectively activated to form a multi-level signal at an output of the circulator or other isolation device. 
     In an embodiment, the driving circuit includes a power amplifier and a switch coupled to an output of the power amplifier, where the switch couples the output of the power amplifier to an RF ground when the power amplifier is in an inactive state. 
     In an embodiment, the transmitter also includes a modulator having a plurality of parallel outputs, where each of the plurality of parallel outputs is coupled to a corresponding input of the plurality of branches. In some embodiments, the modulator is configured to calculate a finite number of first basis functions of a first pulse-width modulated signal based on an input signal, and calculate a finite number of second basis functions of a second pulse-width modulated signal based on the input signal. A first electronic output signal based on the calculated finite number of first basis functions, and a second electronic output signal based on the calculated finite number of second basis functions is formed. The first electronic output signal forms a first of the plurality of parallel outputs, and the second electronic output signal forms a second of the plurality of parallel outputs. The second basis functions are phase shifted from the first basis functions. 
     In an embodiment, the modulator further comprises a digital pulse-width modulator and a digital-to-analog (D/A) converter block coupled to an output of the digital pulse-width modulator. The digital pulse-width modulator is configured to generate a first set of Fourier harmonics approximating a pulse-width modulated signal having a pulse-width proportional to a first digital input signal, and generate a second set of Fourier harmonics approximating a pulse-width modulated signal having a pulse-width proportional to a first digital input signal, where the second set of Fourier harmonics are phase shifted from the first set of Fourier harmonics. The D/A converter block is configured to produce a first output based on the first set of Fourier harmonics and a second output based on the second set of Fourier harmonics. 
     In an embodiment, the modulator is configured to generate a pulse-width modulated signal at each of the plurality of parallel outputs based on modulator input signal. In some embodiments, each of the branches further includes an upconverter coupled between a corresponding output of the modulator and an input of the driving circuit. In an embodiment, each phase shifter includes a 90 degree phase shifter, and each phase shifter has an impedance of about √2 times the first impedance. In some embodiments, the transmitter also includes a bandpass filter coupled to the output port of the circulator or other isolation device. 
     In an embodiment, a system for providing a multi-level pulse-width modulated signal includes a first and second digital pulse-width modulator. The first digital pulse-width modulator is configured to generate a first plurality of basis functions approximating a pulse-width modulated signal having a pulse-width proportional to a first digital input signal, and the second digital pulse-width modulator is configured to generate a second plurality of basis functions approximating a pulse-width modulated signal having a pulse-width proportional to the first digital input signal. The second plurality of basis functions are phase shifted with respect to the first plurality of basis functions. 
     In an embodiment, the first plurality of basis functions includes a first k Fourier Series harmonics of the pulse-width modulated signal. In an embodiment, these first k Fourier Series harmonics are calculated according to: 
                   g   k     ⁢     {           }       =       1     k   ⁢           ⁢   π       ⁢     (       sin   ⁡     (     2   ⁢   π   ⁢           ⁢     kf   P     ⁢   nT     )       -     sin   ⁡     (       2   ⁢   π   ⁢           ⁢     kf   P     ⁢   nT     -     2   ⁢   π   ⁢           ⁢     kx   ⁡     (   nT   )           )         )         ,         
where gk{ } is a time domain representation of a kth basis function, fp is a pulse-width modulation frequency and x(nT) is the input signal. In an embodiment, the system also includes a digital-to-analog converter coupled to an output of the digital pulse-width modulator.
 
     An advantage of embodiments include the ability to operate a multi-level PWM transmitter concept for highly efficient operation which can handle high signal bandwidths and receive bands close to the transmit band without severe interference from the transmit to the receive path. 
     While this invention has been described with reference to illustrative embodiments, this description is not intended to be construed in a limiting sense. Various modifications and combinations of the illustrative embodiments, as well as other embodiments of the invention, will be apparent to persons skilled in the art upon reference to the description. It is therefore intended that the appended claims encompass any such modifications or embodiments.