The Frequency Spectrum of Polar Modulated PWM ... - CiteSeerX
The Frequency Spectrum of Polar Modulated PWM Signals and the Image Problem Shuli Chi
Christian Vogel
Peter Singerl
Signal Processing and Speech Communication Laboratory Graz University of Technology, Austria Email: [email protected]
Telecommunications Research Center Vienna (FTW), Austria Email: [email protected]
Infineon Technologies Austria AG Email: [email protected]
Abstract—Pulse-width modulators (PWMs) are implemented in wireless transmitters to utilize highly efficient switched-mode power amplifiers (SMPAs) such as Class-D,E,F,J amplifiers. This paper gives the analytical expressions for the natural-sampling trailing-edge PWM (TE-NPWM) signals with input signals bounded by [0, 1]. Such PWM modulators can be incorporated in a polar architecture. Important relations of polar modulated PWM signals are analyzed in detail by exploiting the results of the analysis. The image problem is illustrated when a baseband PWM signal is modulated to the passband with the phase-modulated carrier. A properly designed lowpass filter can be used to reduce the image distortions, but increases the total design costs. It further introduces extra delay and has the possible drawback of reducing the overall efficiency. Based on the spectral analysis, two alternative approaches are proposed to show how the image distortions can be reduced without a lowpass filter. The performance of the presented approaches for the image reduction is demonstrated through numerical simulations. Index Terms—Frequency spectrum, PWM, polar modulator, image problem.
I. I NTRODUCTION Conventional RF power amplifiers (PAs) for radio base stations, such as Class-A or Class-AB, are highly linear but inefficient. The low efficiency of PAs results in unnecessary huge power consumption. Many solutions have been proposed to deal with the problem of the poor efficiency, one of which is to utilize inherently nonlinear but efficient SMPAs [1]. High efficiency in SMPAs, however, is obtained when the PA is driven by a constant envelop signal, which only provides the phase information without the amplitude information. Nevertheless, future mobile telecommunication standards such as UMTS or LTE [2] implement QAM and require the amplitude information. Therefore, one solution to efficiently provide the amplitude information as well, is to implement a pulse train modulator before the PA, such that the modulator transforms the amplitude of the complex input signal to a corresponding square-wave signal. The modulator in the baseband processing could be realized by a PWM modulator [3]. The PWM is based on pulses of fixed cycle-time and variable pulse-widths whose average values are directly proportional to the amplitudes of the input signal. The ideal PWM signal has infinite spectral support, which leads to image distortions when the baseband PWM signal is modulated to the passband. The image problem significantly degrades the signal-to-noise ratio (SNR) and the dynamic range (DR) of the modulated output signal. A properly designed lowpass filter can be used to reduce the image distortions, however, it increases the total design costs, introduces extra delay, and has the drawback of efficiency reduction of the SMPA. This paper presents the spectral analysis of TE-NPWM signals with input signals bounded by a(t) ∈ [0, 1]. Such PWM modulators can be incorporated in a polar architecture as shown in Figure 1. Furthermore, important relations of polar modulated PWM signals are analyzed in detail. By exploiting the results of the analysis, we
978-1-4244-8157-6/10/$26.00 ©2010 IEEE 978-1-4244-8156-9/10/$26.00
a(t)
|x(t)|
PWM
x(t)
aP,BB (t) cos(2πfct)
x(t)
cos(φ)
φ(t)
xP,BP (t)
to SMPA
sin(φ) cos(2πfct + φ(t)) Fig. 1.
− sin(2πfct)
Block diagram of a polar PWM modulator.
show how the image distortions can be reduced without a lowpass filter. Two approaches are proposed to tackle the image problem by exploiting a holistic design approach [4], which offer attractive results in the image reduction as well as SNR and DR improvement for a PWM modulator. II. S IGNAL M ODEL A polar PWM modulator [5] is shown in Figure 1. Without loss of generality, it is assumed that the continuous-time input signal x(t) is bounded, i.e., |x(t)| ≤ 1 for all t and is given in the baseband as x(t) = a(t)ejφ(t)
(1)
where a(t) is the amplitude signal and φ(t) is the phase signal. The amplitude signal a(t) is encoded by the baseband TE-NPWM [3] and modulated with the phase-modulated carrier cos(2πfc t + φ(t)). The ideally modulated bandpass signal of (1) can be expressed as xBP (t) = a(t) cos(2πfc t + φ(t)) ˜ (2) ˆ = a(t) cos(φ(t)) cos(2πfc t) − sin(φ(t)) sin(2πfc t) where the index BP denotes the bandpass signal and BB denotes the baseband signal which is used in the following sections. The Fourier transform of (2) is given by ˜ 1ˆ (3) XBP (f ) = X(f − fc ) + X ∗ (−f − fc ) 2 ∗ where X(f ) is the Fourier transform of the signal x(t) and denotes the complex conjugation. The signal XBP (f ) is the ideal spectrum of the modulated signal x(t). III. S PECTRAL A NALYSIS A. Baseband TE-NPWM signal Referring to [3], a generic baseband TE-NPWM signal in Figure 1 is given by
685
aP,BB (t; τ ) =
∞ X
u(t − kTP ) − u(t − kTP − τk )
(4)
k=−∞
ICECS 2010
where the index P denotes the PWM signal. The kth pulse width τk depends on the amplitude of the input signal a(t) and can be expressed as τk = TP a(tk ), where tk denotes the solution to the kth crossing-point equation of the input signal and the PWM reference sawtooth signal. The cycle-time of the PWM pulses is TP = 1/fP and is defined by the period of the reference sawtooth signal. Note that contrast to [3], the input signal a(t) is bounded by 0 ≤ a(t) ≤ 1. Following the same approach as in [3], the spectrum of the TE-NPWM signal can be derived as
Power (dBc)
where A(f ) and An (f ) are the Fourier transforms of a(t) and (a(t))n , respectively. The inverse Fourier transform of (5) produces
B. Bandpass TE-NPWM signal After the baseband signal in (6) is modulated with the phasemodulated carrier, the bandpass PWM signal can be expressed as xP,BP (t) = aP,BB (t) cos(2πfc t + φ(t)) ∞ X 1 ˆ = xBP (t) + sin(2πkfP t)− kπ k=1 ˜ sin(2πkfP t − 2πka(t)) cos(2πfc t + φ(t))
(7)
2
k=1 n=1 n−1
n!
−40 −60 −1
−0.5
0 −20
1 9
x 10
X−P,BP(f)
−40 −60 −80
−1
−0.5
0
0.5
Frequency (Hz)
1 9
x 10
Fig. 2. Frequency-domain illustration of image problems in a polar PWM modulator: (a) illustrates the spectrum of the baseband PWM signal with input a(t). It includes the spectrum of A(f ) and every order of the PWM harmonics. The spacing between adjacent PWM harmonics is equal to the value of fP . (b) illustrates the spectrum of the modulated PWM signal with image problem. The image distortions around the positive carrier are produced by the modulated PWM signal located around the negative carrier, and vice versa.
The spectrum of the convolution of the signal aP,BB (t) with the phase-modulated carrier cos(2πfc t+φ(t)), which can be represented by a negative phase-modulated carrier 12 e−j(2πfc t+φ(t)) and a positive phase-modulated carrier 12 ej(2πfc t+φ(t)) , is shown in Figure 2(b). In order to illustrate the image problem, (8) is expressed as the sum of the negative and positive spectra, which is given by (9)
where according to (3) and (8)
An (f + fc + kfP )+
An (f + fc − kfP ) ∗ Φ∗ (−f )+ (−1) ˆ An (f − fc + kfP )+ o ˜ (−1)n−1 An (f − fc − kfP ) ∗ Φ(f )
0.5
X+P,BP(f)
− XP,BP (f ) =
˜
0
− + (f ) + XP,BP (f ) XP,BP (f ) = XP,BP
and the Fourier transform of (7) is given by XP,BP (f ) = XBP (f )+ ∞ ∞ 1 X X (j2kπ)n−1 nˆ
AP,BB(f)
Frequency (Hz) (b)
(5)
∞ X 1 ˆ aP,BB (t) = a(t) + sin(2πkfP t)− kπ (6) k=1 ˜ sin(2πkfP t − 2πka(t)) P n x where the Taylor series ∞ n=1 x /n! = e − 1 is used. Figure 2(a) illustrates the spectrum of the baseband PWM signal aP,BB (t). It has infinite spectral support, where every order of the PWM harmonics decreases inversely with k. The spacing between adjacent PWM harmonics is equal to the value of fP .
0 −20
−80
Power (dBc)
∞ X ∞ X (j2πk)n−1 ˆ An (f + kfP )+ n! k=1 n=1 ˜ (−1)n−1 An (f − kfP )
AP,BB (f ) = A(f ) +
(a)
ˆ
∞ ∞ 1 ∗ 1 X X (j2πk)n−1 X (−f − fc ) + × 2 2 k=1 n=1 n!
An (f + fc + kfP )+
(10)
˜ (−1)n−1 An (f + fc − kfP ) ∗ Φ∗ (−f )
(8) and
where Φ(f ) and Φ∗ (−f ) are the Fourier transforms of the phase signals ejφ(t) and e−jφ(t) , respectively. The signal XP,BP (f ) includes the spectrum XBP (f ) of the modulated input signal in (3) and the PWM harmonic distortions. IV. I MAGE R EDUCTION In the following, we will explain the spectral analysis to propose two methods avoiding the necessity of a lowpass filter after the PWM. The image problem is caused by the infinitely large bandwidth of the PWM signal aP,BB (t) when it is modulated with the phasemodulated carrier. Figure 2(a) shows the frequency spectrum of the baseband PWM signal aP,BB (t), where the main power of the input signal a(t) is located around the zero frequency. The PWM harmonics are located at the frequencies that are multiples of the PWM frequency fP and the power of the kth harmonic decreases inversely with k with reference to (6).
+ (f ) = XP,BP
ˆ
∞ ∞ 1 1 X X (j2πk)n−1 X(f − fc ) + × 2 2 k=1 n=1 n!
An (f − fc + kfP )+
(11)
˜ (−1)n−1 An (f − fc − kfP ) ∗ Φ(f ). − + (f ) and XP,BP (f ) are depicted The spectra of the signals XP,BP as grey and black lines in Figure 2(b), respectively. The image distortions around the positive carrier are produced by the modulated PWM signal located around the negative carrier, and vice versa. The image distortions can be reduced by exploiting the image gap to avoid the image peak that is located around the carrier frequency or by eliminating the image harmonic that is located around the carrier frequency.
A. Exploiting of the image gap From Figure 2(b) we see that, if the desired signal (black) located around the carrier frequency is not superposed with the image
686
harmonic (grey), but in the middle of the adjacent image harmonics, the peak of the image distortions can be avoided. This can be done by properly “shifting” the image harmonics, which can be realized by changing the PWM frequency to a proper value. First of all, the order of the image harmonic located around the carrier frequency (the qth harmonic) can be computed as « „ 2fc (12) q = round fP
where Q(·) is a quantizer function and Δk = 1/k is the quantization interval parameter. The resulting a ˆ(t) is a discrete-amplitude continous-time signal. The qth image harmonic, given in (12), which introduces distortions in the vicinity of the desired signal needs to be eliminated. With k = q in (17), the qth image harmonic is eliminated if „ « a(t) (18) a ˆ(t) = Δq · round Δq
where q is an integer and relies on the desired value of the PWM frequency fP , which has to be carefully determined. On the one hand, PWM frequency is affected by the bandwidth of the input signal x(t) and with a larger PWM frequency it is easier to perform the bandpass filtering in the last stage of the transmitter (filters the output of the SMPA). On the other hand, the efficiency of the SMPA is higher with a smaller PWM frequency. When the signal is located in the middle of two adjacent image harmonics, from Figure 2(b) and (8), a special relation between the carrier frequency fc and the new PWM frequency fP,gap by exploiting the image gap can be derived as
and Δq = 1/q is the quantization interval parameter. Additionally, because of the 2π periodicity in (16), the l · kth harmonic for l ∈ Z will be eliminated. However, as the signal a(t) gets quantized a quantization noise is introduced.
−fc + qfP,gap +
1 fP,gap = fc 2
(13)
and fp,gap can be expressed as fP,gap =
2fc . q + 12
(14)
For example, with an input signal of 5 MHz bandwidth, a PWM frequency of 50 MHz and a carrier frequency of 1 GHz, q is given with (12) by q = 2 × 1GHz/50MHz = 40 and fP,gap is calculated with (14) as 49.4 MHz. By this approach, the modulated input signal around the carrier is located in the middle of two adjacent image harmonics and the image peak distortions can be avoided. B. Elimination of the image distortion Instead of “shifting” the harmonics by changing the PWM frequency, an alternative approach is proposed to eliminate the image harmonic located around the desired signal by quantizing the input signal a(t). The image distortions around the desired signal are produced by the second term in (10) and given by −r (f ) = XP,BP
∞ ∞ 1 X X (j2πk)n−1 (−1)n−1 × 2 k=1 n=1 n!
(15)
An (f + fc − kfP ) ∗ Φ∗ (−f ). ∞ ∞ 1 X X (−j2πk)n−1 (a(t))n × 2 n! n=1 k=1
e−j(2πfc t−2πkfP t+φ(t)) ∞ X 1 (1 − e−j2πka(t) )× = j4πk
In this section the performance of the approaches considered in the last section are investigated. The carrier frequency fc is set to 1 GHz and the PWM signal is created by a discrete-multitone (DMT) signal with a bandwidth Bs of 5 MHz and a peak to average power ratio (PAPR) of 7 dB. The PWM frequency fP is set to 50 MHz for the simulations. The simulation time step is set to 1/(64GHz) to capture the outoff-band spectrum caused by the upconversion of the large bandwidth of the PWM signal. Due to the finite time resolution the simulations introduce quantization noise that degrades the SNR and DR. The normalized in-band dynamic range DRin is used to measure the performance of the image reduction, which is defined as h i1 0 + (f + fc )|2 maxf ∈fin |XP,BP A h i (19) DRin = 10 log10 @ maxf ∈fin |Din (f )|2 + where XP,BP (f + fc ) is the spectrum of the demodulated signal of aP,BB (t) modulated by the positive phase-modulated carrier. The frequency f ∈ fin denotes that f is bounded by -Bs /2 MHz ≤ f ≤ Bs /2 MHz, which is the same as performing an ideal lowpass filter with a bandwidth Bs . The spectrum Din (f ) consists of the demodulated image harmonics filtered by the same ideal lowpass filter. Additionally, two other measurements are presented to give a view of an overall performance of the PWM modulator with the image reduction. First, the in-band SNR is defined in frequency-domain as « „ Ef ∈fin [X 2 (f )] (20) SNR = 10 log10 Ef ∈fin [e2 (f )]
where E[·] denotes the mean value. The signal X(f ) is the Fourier transform of the input signal x(t) and the error signal e(f ) is given by (21) e(f ) = XP,BP (f + fc ) − X(f )
The inverse Fourier transform of (15) is given by x−r P,BP (t) =
V. S IMULATION R ESULTS
(16)
k=1
e−j(2πfc t−2πkfP t+φ(t)) and shows the image harmonics that influence the desired signal. From (16) it can be seen that the kth image harmonic is eliminated if 1−e−j2kπa(t) = 0, which leads to a quantized version of the input signal a(t) as „ « a(t) (17) a ˆ(t) = Q (a(t)) = Δk · round Δk
where XP,BP (f + fc ) is the spetrum of the demodulated signal of aP,BB (t) modulated by the phase-modulated carrier cos(2πfc t + φ(t)). Note that XP,BP (f + fc ) is the signal including the image + (f + fc ). components which are not included in XP,BP Second, we define the normalized out-of-band dynamic range DR as h i1 0 maxf ∈fin |XP,BP (f + fc )|2 h i A (22) DR = 10 log10 @ maxf ∈fout |Dout (f )|2 where f ∈ fout denotes that f is bounded by Bs /2 MHz ≤ |f | ≤ 2Bs MHz. The spectrum Dout (f ) consists of the demodulated outof-band distortion within the bandwidth of fout .
687
0
0
−10
−10
DRin
DR −20
Power (dBc)
Power (dBc)
−20 −30 −40 −50 −60
−80 0.85
0.95
1
1.05
1.1
Frequency (Hz)
−80 0.85
1.15
+
XP,BP(f) X−P,BP(f) 0.9
0.95
1
1.05
Frequency (Hz)
9
x 10
Fig. 3. Spectrum of the uncompensated modulated PWM signal with the image problem.
1.1
1.15 9
x 10
Fig. 5. Spectrum of the modulated PWM signal by the elimination of the image distortion approach.
Figure 5 shows the spectrum by the elimination of the image distortion approach with q = 40, fp = 50 MHz, where DRin , SNR and DR are 65.8 dB, 46.0 dB and 53.4 dB, which gives approximately 28.2 dB, 5.3 dB and 5.0 dB improvement, respectively.
0 −10
Power (dBc)
−50
−70
X−P,BP(f) 0.9
−40
−60
+
XP,BP(f)
−70
−30
−20
VI. C ONCLUSION
−30
In this paper, the time- and frequency-domain representations of TE-NPWM signals with input signals bounded by a(t) ∈ [0, 1] are presented. Such PWM modulators can be incorporated in a polar architecture. The polar modulated PWM signals are analyzed mathematically. The image problem is illustrated when a baseband PWM signal is modulated to the passband with the phase-modulated carrier. With the presented mathematical analysis of a polar modulator, the image distortions can be reduced without a lowpass filter. Two approaches are given to reduce the image distortions. The simulation results show that the presented approaches have a good performance in image reduction as well as in SNR and DR improvement.
−40 −50 −60
X+P,BP(f)
−70
X−P,BP(f)
−80 0.85
0.9
0.95
1
1.05
Frequency (Hz)
1.1
1.15 9
x 10
Fig. 4. Spectrum of the modulated PWM signal by the exploiting of the image gap approach.
These two methods consider the distortions that include the image distortions as well as other distortions such as quantization noise. The SNR and DR are given as references to see the overall performance of the PWM modulator with the image reduction approaches. Figure 3 shows the spectrum of the uncompensated output signal with the image problem. The image harmonic that introduces the image distortions is located around the carrier frequency and significantly reduces the DRin , SNR and DR down to 37.6 dB, 40.7 dB and 48.4 dB, respectively. Figure 4 shows the spectrum after exploiting the image gap approach with q = 40 and fP,gap = 49.4 MHz. It can be seen that the desired signal is located in the middle of two neighboring harmonics to avoid the image peak distortions. By this approach, DRin , SNR and DR are 59.3 dB, 50.5 dB and 58.6 dB, which leads to approximately 21.7 dB, 9.8 dB and 10.2 dB improvement, respectively.
ACKNOWLEDGMENTS Shuli Chi was supported by the Zukunftsfonds Steiermark/Land Steiermark in the context of the project GreenPArk with the project No. 5061. R EFERENCES [1] S. C. Cripps, “RF Power Amplifiers for Wireless Communications,” Artech House Microwave Library, 2rd ed, 2006. [2] H. Ekstrom, A. Furuskar, J. Karlsson, M. Meyer, S. Parkvall, J. Torsner and M. Wahlqvist, “Technical solutions for the 3G long-term evolution,” IEEE Communications Magazine, vol. 44, no. 3, pp. 38-45, March 2006. [3] Z. Song and D. V. Sarwate, “The frequency spectrum of pulse width modulated signals,” Signal Processing, vol. 83, no. 10, pp. 2227-2258, October 2003. [4] F. Dielacher, C. Vogel, P. Singerl, S. Mendel and A. Wiesbauer, “A Holistic Design Approach for Systems on Chip”, 22nd IEEE International SOC Conference (SOCC 2009), Belfast (Northern Irland, UK), 9-11 September 2009, pp. 301-306. [5] J. Groe, “Polar Transmitters for Wireless Communications,” IEEE Communications Magazine, vol. 45, no. 9, pp. 58-63, September 2007 [6] A. V. Oppenheim, R. W. Schafer and J. R. Buck, “Discrete-time signal processing,” Prentice Hall, 2rd ed, 1999.
688
Christian Vogel
Peter Singerl
Signal Processing and Speech Communication Laboratory Graz University of Technology, Austria Email: [email protected]
Telecommunications Research Center Vienna (FTW), Austria Email: [email protected]
Infineon Technologies Austria AG Email: [email protected]
Abstract—Pulse-width modulators (PWMs) are implemented in wireless transmitters to utilize highly efficient switched-mode power amplifiers (SMPAs) such as Class-D,E,F,J amplifiers. This paper gives the analytical expressions for the natural-sampling trailing-edge PWM (TE-NPWM) signals with input signals bounded by [0, 1]. Such PWM modulators can be incorporated in a polar architecture. Important relations of polar modulated PWM signals are analyzed in detail by exploiting the results of the analysis. The image problem is illustrated when a baseband PWM signal is modulated to the passband with the phase-modulated carrier. A properly designed lowpass filter can be used to reduce the image distortions, but increases the total design costs. It further introduces extra delay and has the possible drawback of reducing the overall efficiency. Based on the spectral analysis, two alternative approaches are proposed to show how the image distortions can be reduced without a lowpass filter. The performance of the presented approaches for the image reduction is demonstrated through numerical simulations. Index Terms—Frequency spectrum, PWM, polar modulator, image problem.
I. I NTRODUCTION Conventional RF power amplifiers (PAs) for radio base stations, such as Class-A or Class-AB, are highly linear but inefficient. The low efficiency of PAs results in unnecessary huge power consumption. Many solutions have been proposed to deal with the problem of the poor efficiency, one of which is to utilize inherently nonlinear but efficient SMPAs [1]. High efficiency in SMPAs, however, is obtained when the PA is driven by a constant envelop signal, which only provides the phase information without the amplitude information. Nevertheless, future mobile telecommunication standards such as UMTS or LTE [2] implement QAM and require the amplitude information. Therefore, one solution to efficiently provide the amplitude information as well, is to implement a pulse train modulator before the PA, such that the modulator transforms the amplitude of the complex input signal to a corresponding square-wave signal. The modulator in the baseband processing could be realized by a PWM modulator [3]. The PWM is based on pulses of fixed cycle-time and variable pulse-widths whose average values are directly proportional to the amplitudes of the input signal. The ideal PWM signal has infinite spectral support, which leads to image distortions when the baseband PWM signal is modulated to the passband. The image problem significantly degrades the signal-to-noise ratio (SNR) and the dynamic range (DR) of the modulated output signal. A properly designed lowpass filter can be used to reduce the image distortions, however, it increases the total design costs, introduces extra delay, and has the drawback of efficiency reduction of the SMPA. This paper presents the spectral analysis of TE-NPWM signals with input signals bounded by a(t) ∈ [0, 1]. Such PWM modulators can be incorporated in a polar architecture as shown in Figure 1. Furthermore, important relations of polar modulated PWM signals are analyzed in detail. By exploiting the results of the analysis, we
978-1-4244-8157-6/10/$26.00 ©2010 IEEE 978-1-4244-8156-9/10/$26.00
a(t)
|x(t)|
PWM
x(t)
aP,BB (t) cos(2πfct)
x(t)
cos(φ)
φ(t)
xP,BP (t)
to SMPA
sin(φ) cos(2πfct + φ(t)) Fig. 1.
− sin(2πfct)
Block diagram of a polar PWM modulator.
show how the image distortions can be reduced without a lowpass filter. Two approaches are proposed to tackle the image problem by exploiting a holistic design approach [4], which offer attractive results in the image reduction as well as SNR and DR improvement for a PWM modulator. II. S IGNAL M ODEL A polar PWM modulator [5] is shown in Figure 1. Without loss of generality, it is assumed that the continuous-time input signal x(t) is bounded, i.e., |x(t)| ≤ 1 for all t and is given in the baseband as x(t) = a(t)ejφ(t)
(1)
where a(t) is the amplitude signal and φ(t) is the phase signal. The amplitude signal a(t) is encoded by the baseband TE-NPWM [3] and modulated with the phase-modulated carrier cos(2πfc t + φ(t)). The ideally modulated bandpass signal of (1) can be expressed as xBP (t) = a(t) cos(2πfc t + φ(t)) ˜ (2) ˆ = a(t) cos(φ(t)) cos(2πfc t) − sin(φ(t)) sin(2πfc t) where the index BP denotes the bandpass signal and BB denotes the baseband signal which is used in the following sections. The Fourier transform of (2) is given by ˜ 1ˆ (3) XBP (f ) = X(f − fc ) + X ∗ (−f − fc ) 2 ∗ where X(f ) is the Fourier transform of the signal x(t) and denotes the complex conjugation. The signal XBP (f ) is the ideal spectrum of the modulated signal x(t). III. S PECTRAL A NALYSIS A. Baseband TE-NPWM signal Referring to [3], a generic baseband TE-NPWM signal in Figure 1 is given by
685
aP,BB (t; τ ) =
∞ X
u(t − kTP ) − u(t − kTP − τk )
(4)
k=−∞
ICECS 2010
where the index P denotes the PWM signal. The kth pulse width τk depends on the amplitude of the input signal a(t) and can be expressed as τk = TP a(tk ), where tk denotes the solution to the kth crossing-point equation of the input signal and the PWM reference sawtooth signal. The cycle-time of the PWM pulses is TP = 1/fP and is defined by the period of the reference sawtooth signal. Note that contrast to [3], the input signal a(t) is bounded by 0 ≤ a(t) ≤ 1. Following the same approach as in [3], the spectrum of the TE-NPWM signal can be derived as
Power (dBc)
where A(f ) and An (f ) are the Fourier transforms of a(t) and (a(t))n , respectively. The inverse Fourier transform of (5) produces
B. Bandpass TE-NPWM signal After the baseband signal in (6) is modulated with the phasemodulated carrier, the bandpass PWM signal can be expressed as xP,BP (t) = aP,BB (t) cos(2πfc t + φ(t)) ∞ X 1 ˆ = xBP (t) + sin(2πkfP t)− kπ k=1 ˜ sin(2πkfP t − 2πka(t)) cos(2πfc t + φ(t))
(7)
2
k=1 n=1 n−1
n!
−40 −60 −1
−0.5
0 −20
1 9
x 10
X−P,BP(f)
−40 −60 −80
−1
−0.5
0
0.5
Frequency (Hz)
1 9
x 10
Fig. 2. Frequency-domain illustration of image problems in a polar PWM modulator: (a) illustrates the spectrum of the baseband PWM signal with input a(t). It includes the spectrum of A(f ) and every order of the PWM harmonics. The spacing between adjacent PWM harmonics is equal to the value of fP . (b) illustrates the spectrum of the modulated PWM signal with image problem. The image distortions around the positive carrier are produced by the modulated PWM signal located around the negative carrier, and vice versa.
The spectrum of the convolution of the signal aP,BB (t) with the phase-modulated carrier cos(2πfc t+φ(t)), which can be represented by a negative phase-modulated carrier 12 e−j(2πfc t+φ(t)) and a positive phase-modulated carrier 12 ej(2πfc t+φ(t)) , is shown in Figure 2(b). In order to illustrate the image problem, (8) is expressed as the sum of the negative and positive spectra, which is given by (9)
where according to (3) and (8)
An (f + fc + kfP )+
An (f + fc − kfP ) ∗ Φ∗ (−f )+ (−1) ˆ An (f − fc + kfP )+ o ˜ (−1)n−1 An (f − fc − kfP ) ∗ Φ(f )
0.5
X+P,BP(f)
− XP,BP (f ) =
˜
0
− + (f ) + XP,BP (f ) XP,BP (f ) = XP,BP
and the Fourier transform of (7) is given by XP,BP (f ) = XBP (f )+ ∞ ∞ 1 X X (j2kπ)n−1 nˆ
AP,BB(f)
Frequency (Hz) (b)
(5)
∞ X 1 ˆ aP,BB (t) = a(t) + sin(2πkfP t)− kπ (6) k=1 ˜ sin(2πkfP t − 2πka(t)) P n x where the Taylor series ∞ n=1 x /n! = e − 1 is used. Figure 2(a) illustrates the spectrum of the baseband PWM signal aP,BB (t). It has infinite spectral support, where every order of the PWM harmonics decreases inversely with k. The spacing between adjacent PWM harmonics is equal to the value of fP .
0 −20
−80
Power (dBc)
∞ X ∞ X (j2πk)n−1 ˆ An (f + kfP )+ n! k=1 n=1 ˜ (−1)n−1 An (f − kfP )
AP,BB (f ) = A(f ) +
(a)
ˆ
∞ ∞ 1 ∗ 1 X X (j2πk)n−1 X (−f − fc ) + × 2 2 k=1 n=1 n!
An (f + fc + kfP )+
(10)
˜ (−1)n−1 An (f + fc − kfP ) ∗ Φ∗ (−f )
(8) and
where Φ(f ) and Φ∗ (−f ) are the Fourier transforms of the phase signals ejφ(t) and e−jφ(t) , respectively. The signal XP,BP (f ) includes the spectrum XBP (f ) of the modulated input signal in (3) and the PWM harmonic distortions. IV. I MAGE R EDUCTION In the following, we will explain the spectral analysis to propose two methods avoiding the necessity of a lowpass filter after the PWM. The image problem is caused by the infinitely large bandwidth of the PWM signal aP,BB (t) when it is modulated with the phasemodulated carrier. Figure 2(a) shows the frequency spectrum of the baseband PWM signal aP,BB (t), where the main power of the input signal a(t) is located around the zero frequency. The PWM harmonics are located at the frequencies that are multiples of the PWM frequency fP and the power of the kth harmonic decreases inversely with k with reference to (6).
+ (f ) = XP,BP
ˆ
∞ ∞ 1 1 X X (j2πk)n−1 X(f − fc ) + × 2 2 k=1 n=1 n!
An (f − fc + kfP )+
(11)
˜ (−1)n−1 An (f − fc − kfP ) ∗ Φ(f ). − + (f ) and XP,BP (f ) are depicted The spectra of the signals XP,BP as grey and black lines in Figure 2(b), respectively. The image distortions around the positive carrier are produced by the modulated PWM signal located around the negative carrier, and vice versa. The image distortions can be reduced by exploiting the image gap to avoid the image peak that is located around the carrier frequency or by eliminating the image harmonic that is located around the carrier frequency.
A. Exploiting of the image gap From Figure 2(b) we see that, if the desired signal (black) located around the carrier frequency is not superposed with the image
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harmonic (grey), but in the middle of the adjacent image harmonics, the peak of the image distortions can be avoided. This can be done by properly “shifting” the image harmonics, which can be realized by changing the PWM frequency to a proper value. First of all, the order of the image harmonic located around the carrier frequency (the qth harmonic) can be computed as « „ 2fc (12) q = round fP
where Q(·) is a quantizer function and Δk = 1/k is the quantization interval parameter. The resulting a ˆ(t) is a discrete-amplitude continous-time signal. The qth image harmonic, given in (12), which introduces distortions in the vicinity of the desired signal needs to be eliminated. With k = q in (17), the qth image harmonic is eliminated if „ « a(t) (18) a ˆ(t) = Δq · round Δq
where q is an integer and relies on the desired value of the PWM frequency fP , which has to be carefully determined. On the one hand, PWM frequency is affected by the bandwidth of the input signal x(t) and with a larger PWM frequency it is easier to perform the bandpass filtering in the last stage of the transmitter (filters the output of the SMPA). On the other hand, the efficiency of the SMPA is higher with a smaller PWM frequency. When the signal is located in the middle of two adjacent image harmonics, from Figure 2(b) and (8), a special relation between the carrier frequency fc and the new PWM frequency fP,gap by exploiting the image gap can be derived as
and Δq = 1/q is the quantization interval parameter. Additionally, because of the 2π periodicity in (16), the l · kth harmonic for l ∈ Z will be eliminated. However, as the signal a(t) gets quantized a quantization noise is introduced.
−fc + qfP,gap +
1 fP,gap = fc 2
(13)
and fp,gap can be expressed as fP,gap =
2fc . q + 12
(14)
For example, with an input signal of 5 MHz bandwidth, a PWM frequency of 50 MHz and a carrier frequency of 1 GHz, q is given with (12) by q = 2 × 1GHz/50MHz = 40 and fP,gap is calculated with (14) as 49.4 MHz. By this approach, the modulated input signal around the carrier is located in the middle of two adjacent image harmonics and the image peak distortions can be avoided. B. Elimination of the image distortion Instead of “shifting” the harmonics by changing the PWM frequency, an alternative approach is proposed to eliminate the image harmonic located around the desired signal by quantizing the input signal a(t). The image distortions around the desired signal are produced by the second term in (10) and given by −r (f ) = XP,BP
∞ ∞ 1 X X (j2πk)n−1 (−1)n−1 × 2 k=1 n=1 n!
(15)
An (f + fc − kfP ) ∗ Φ∗ (−f ). ∞ ∞ 1 X X (−j2πk)n−1 (a(t))n × 2 n! n=1 k=1
e−j(2πfc t−2πkfP t+φ(t)) ∞ X 1 (1 − e−j2πka(t) )× = j4πk
In this section the performance of the approaches considered in the last section are investigated. The carrier frequency fc is set to 1 GHz and the PWM signal is created by a discrete-multitone (DMT) signal with a bandwidth Bs of 5 MHz and a peak to average power ratio (PAPR) of 7 dB. The PWM frequency fP is set to 50 MHz for the simulations. The simulation time step is set to 1/(64GHz) to capture the outoff-band spectrum caused by the upconversion of the large bandwidth of the PWM signal. Due to the finite time resolution the simulations introduce quantization noise that degrades the SNR and DR. The normalized in-band dynamic range DRin is used to measure the performance of the image reduction, which is defined as h i1 0 + (f + fc )|2 maxf ∈fin |XP,BP A h i (19) DRin = 10 log10 @ maxf ∈fin |Din (f )|2 + where XP,BP (f + fc ) is the spectrum of the demodulated signal of aP,BB (t) modulated by the positive phase-modulated carrier. The frequency f ∈ fin denotes that f is bounded by -Bs /2 MHz ≤ f ≤ Bs /2 MHz, which is the same as performing an ideal lowpass filter with a bandwidth Bs . The spectrum Din (f ) consists of the demodulated image harmonics filtered by the same ideal lowpass filter. Additionally, two other measurements are presented to give a view of an overall performance of the PWM modulator with the image reduction. First, the in-band SNR is defined in frequency-domain as « „ Ef ∈fin [X 2 (f )] (20) SNR = 10 log10 Ef ∈fin [e2 (f )]
where E[·] denotes the mean value. The signal X(f ) is the Fourier transform of the input signal x(t) and the error signal e(f ) is given by (21) e(f ) = XP,BP (f + fc ) − X(f )
The inverse Fourier transform of (15) is given by x−r P,BP (t) =
V. S IMULATION R ESULTS
(16)
k=1
e−j(2πfc t−2πkfP t+φ(t)) and shows the image harmonics that influence the desired signal. From (16) it can be seen that the kth image harmonic is eliminated if 1−e−j2kπa(t) = 0, which leads to a quantized version of the input signal a(t) as „ « a(t) (17) a ˆ(t) = Q (a(t)) = Δk · round Δk
where XP,BP (f + fc ) is the spetrum of the demodulated signal of aP,BB (t) modulated by the phase-modulated carrier cos(2πfc t + φ(t)). Note that XP,BP (f + fc ) is the signal including the image + (f + fc ). components which are not included in XP,BP Second, we define the normalized out-of-band dynamic range DR as h i1 0 maxf ∈fin |XP,BP (f + fc )|2 h i A (22) DR = 10 log10 @ maxf ∈fout |Dout (f )|2 where f ∈ fout denotes that f is bounded by Bs /2 MHz ≤ |f | ≤ 2Bs MHz. The spectrum Dout (f ) consists of the demodulated outof-band distortion within the bandwidth of fout .
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0
0
−10
−10
DRin
DR −20
Power (dBc)
Power (dBc)
−20 −30 −40 −50 −60
−80 0.85
0.95
1
1.05
1.1
Frequency (Hz)
−80 0.85
1.15
+
XP,BP(f) X−P,BP(f) 0.9
0.95
1
1.05
Frequency (Hz)
9
x 10
Fig. 3. Spectrum of the uncompensated modulated PWM signal with the image problem.
1.1
1.15 9
x 10
Fig. 5. Spectrum of the modulated PWM signal by the elimination of the image distortion approach.
Figure 5 shows the spectrum by the elimination of the image distortion approach with q = 40, fp = 50 MHz, where DRin , SNR and DR are 65.8 dB, 46.0 dB and 53.4 dB, which gives approximately 28.2 dB, 5.3 dB and 5.0 dB improvement, respectively.
0 −10
Power (dBc)
−50
−70
X−P,BP(f) 0.9
−40
−60
+
XP,BP(f)
−70
−30
−20
VI. C ONCLUSION
−30
In this paper, the time- and frequency-domain representations of TE-NPWM signals with input signals bounded by a(t) ∈ [0, 1] are presented. Such PWM modulators can be incorporated in a polar architecture. The polar modulated PWM signals are analyzed mathematically. The image problem is illustrated when a baseband PWM signal is modulated to the passband with the phase-modulated carrier. With the presented mathematical analysis of a polar modulator, the image distortions can be reduced without a lowpass filter. Two approaches are given to reduce the image distortions. The simulation results show that the presented approaches have a good performance in image reduction as well as in SNR and DR improvement.
−40 −50 −60
X+P,BP(f)
−70
X−P,BP(f)
−80 0.85
0.9
0.95
1
1.05
Frequency (Hz)
1.1
1.15 9
x 10
Fig. 4. Spectrum of the modulated PWM signal by the exploiting of the image gap approach.
These two methods consider the distortions that include the image distortions as well as other distortions such as quantization noise. The SNR and DR are given as references to see the overall performance of the PWM modulator with the image reduction approaches. Figure 3 shows the spectrum of the uncompensated output signal with the image problem. The image harmonic that introduces the image distortions is located around the carrier frequency and significantly reduces the DRin , SNR and DR down to 37.6 dB, 40.7 dB and 48.4 dB, respectively. Figure 4 shows the spectrum after exploiting the image gap approach with q = 40 and fP,gap = 49.4 MHz. It can be seen that the desired signal is located in the middle of two neighboring harmonics to avoid the image peak distortions. By this approach, DRin , SNR and DR are 59.3 dB, 50.5 dB and 58.6 dB, which leads to approximately 21.7 dB, 9.8 dB and 10.2 dB improvement, respectively.
ACKNOWLEDGMENTS Shuli Chi was supported by the Zukunftsfonds Steiermark/Land Steiermark in the context of the project GreenPArk with the project No. 5061. R EFERENCES [1] S. C. Cripps, “RF Power Amplifiers for Wireless Communications,” Artech House Microwave Library, 2rd ed, 2006. [2] H. Ekstrom, A. Furuskar, J. Karlsson, M. Meyer, S. Parkvall, J. Torsner and M. Wahlqvist, “Technical solutions for the 3G long-term evolution,” IEEE Communications Magazine, vol. 44, no. 3, pp. 38-45, March 2006. [3] Z. Song and D. V. Sarwate, “The frequency spectrum of pulse width modulated signals,” Signal Processing, vol. 83, no. 10, pp. 2227-2258, October 2003. [4] F. Dielacher, C. Vogel, P. Singerl, S. Mendel and A. Wiesbauer, “A Holistic Design Approach for Systems on Chip”, 22nd IEEE International SOC Conference (SOCC 2009), Belfast (Northern Irland, UK), 9-11 September 2009, pp. 301-306. [5] J. Groe, “Polar Transmitters for Wireless Communications,” IEEE Communications Magazine, vol. 45, no. 9, pp. 58-63, September 2007 [6] A. V. Oppenheim, R. W. Schafer and J. R. Buck, “Discrete-time signal processing,” Prentice Hall, 2rd ed, 1999.
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