Chapter 3

67

Part I Chapter 3

Digital Pulse Modulation The continuing expansion of digital techniques in the field of audio rises a question – is it possible to convert the digital encoded signals (PCM) directly to a pulse modulated signal for subsequent power amplification? The motivation is of course the topological simplification in both the digital to analog conversion stage and the subsequent power amplification stage. Intuitively, it is advantageous to keep the signal digital “as long as possible” with the accuracy and rigidity that generally follows. However, fundamental problems have persisted within digital PMAs although the field has attracted significant attention within the last decade. The topic of this chapter is optimal digital modulator realization dedicated to digital PMAs. The known methods are reviewed to determine the optimal methods on the performance / complexity scale. Previous research is extended by a fundamental and general spectral analysis of digital pulse modulation methods. A simple digital PWM modulator design methodology is presented. The methodology can be used for systematic design of digital pulse modulators based on fundamental specifications for harmonic distortion and dynamic range.

3.1

The digital PMA paradox

Fig. 3.1 shows the basic digital PMA system. Compared to a DAC/Analog PMA system, the digital PMA offers some simplifications in that the post section of the DAC can be simplified. Furthermore, no analog pulse modulator is needed. Within the last decade, much progress on digital pulse modulation methods for digital PMA systems has been

68

Digital Pulse Modulation

PCM Input

Digital Modulator

Switching Power Stage

Filter

Fig. 3.1 Digital PMA topology

shown [Sa86], [Le91], [Go91], [Go91], [Sh92], [Hi94] and several methods have been developed that provide a high level of modulator performance. With the obvious topological advantages that are offered by the digital PMA, it might seem paradoxical that the topology has not already found widespread use in commercial products. The answer lies in the general misconception that the digital PMA inherently offers improved performance by keeping the signal digital and pulse modulated throughout the audio chain. The signal enters the analog domain at the modulator output as an analog pulse signal. Only the digital modulator is in effect digital with the accuracy and rigidity the follows, whereas both of the following elements are inherently analog and non-linear elements. Hence, the modulated signal is sensitive to pulse jitter and amplitude distortion. It is bound with considerable difficulty to maintain a reasonable performance level throughout the subsequent power amplification and demodulation stages. For robust power amplification, it is necessary with compensation for the distortion introduced during power amplification. The non-linear characteristic of the power stage is analyzed more closely in Chapter 4, and the important issue of optimal power conversion with inherent error correction for digital PMA systems, is the topic of Chapter 9. This chapter exclusively focuses on digital modulator realization for digital PMAs.

3.2

Digital Pulse modulation methods

The digital pulse modulator can be realized by utilizing both digital PWM and digital PDM. The properties of the two methods in digital PMA applications are discussed shortly in the following.

3.2.1 Digital PDM Digital PDM modulators have been extensively researched as e.g. in [Na87], [Ad90], [Ha91], [Ad91], [Ri94] and found widespread use in commercial DAC’s and ADC’s within the last decade due to the inherent advantage of PDM on a performance/complexity scale. The digital PDM modulator topology, shown in Fig. 3.2, may be used directly as modulator in the digital PMA system in Fig. 3.1. The noise-shaping filter serves to shape the quantization noise such that the base band performance can be maintained. PDM has the advantage over PWM in that it can realize virtually distortion free performance over the base band with proper selection of oversampling ratio and loop filter parameters. Another advantage is the low complexity - the modulator is realized by a linear digital filter and a quantizer. Intuitively PDM is highly “digital” by the on/off characteristic of each pulse whereas PWM codes the information into a pulse width. On the other hand, a range of problems exists with digital PDM modulators in PMA applications, although the rigidity of the digital domain allows implementation of higher order modulators. However, even with a loop filter between fourth and eighth order the resulting carrier frequency will be high. Thus, for a fourth order filter the resulting sampling frequency is 2.82MHz for reasonable audio performance [Kl97]. This leads to an average pulse switching frequency which is dependent on the audio signal but will have an average switching frequency about

Chapter 3

69 .

x(n)

b fS

Interpolation

brq

H(z)

I fS

I fS

Loop filter

y(n)

Quantizer

Fig. 3.2 General digital PDM modulator topology

800KHz-1MHz. Reasonable compromises between filter order and oversampling ration are 64x oversampling / fourth order filter [Kl97] or 32x oversampling / eighth order filter [Sm94]. A side effect of higher order modulators is the limits the modulation depth which becomes a problem in PMA applications, since this increases the power supply rail voltage for a given output power. A higher power supply voltage will compromise both performance and efficiency. There has been some activity in recent years to solve the problem of a high idling pulse activity in PDM modulators for digital PMAs. An interesting approach is presented in [Ma95] where the individual output bits of the modulator are inverted in a controlled fashion in order to minimize the resulting pulse frequency (especially at idle). By applying the pulse inversion within the loop, the distortion arising from such a modification is controlled. Some improvements over conventional PDM have been shown in [Ma95] using a higher order loop filter (7. order). However, the improvements come with an considerable increase in implementation complexity and stability issues constrains the modulation depth to below 0.5 which is very problematic with a switching power output stage. A familiar system is presented in [Ma96] where a lower carrier frequency is obtained by grouping together output pulses, which is modeled as a linear filtering, decimation and pulse width modulation process. The method attempts to emulate PWM. However, the properties and inherent limitations remain the same. To conclude on PDM, no method has yet been presented that can match previously presented results on digital pulse width modulation. Consequently, PDM and its variants will notbe considered further.

3.2.2 Digital PWM The practical conversion of a digital PCM signal to a uniformly sampled pulse width modulated signal is remarkable simple. Fig. 3.3 shows an example system that converts the b bit represented input to a UPWM signal at the carrier rate f c equal to the sample rate f s of the PCM signal. The digital modulator uses a high frequency b bit counter to define the timing edges. It is essential, that the conversion from PCM to UPWM is realized without loss of information. Hence, the precision of the bit clock is critical for maintaining system performance and good long-term stability and phase noise and other elements causing jitter

Sample clock: fS

CLK

S

Load Data

Parallel input (b bit)

Down TC Counter CLK

Bit clock: fS 2b

R

SR Latch CLK

DATA

Out

X1

TC

fC =fS

Out

Fig. 3.3 Basic digital PCM – PWM conversion.

X2

70

Digital Pulse Modulation

Sampling method

Edge Single sided

Levels Abbreviation Two (AD) UADS Uniform sampling Three (BD) UBDS (UPWM) Double sided Two (AD) UADD Three (BD) UBDD Table 3.1 Fundamental uniformly sampled PWM (UPWM) methods.

have to be controlled. The requirement for counting speed is a fundamental limitation in digital PWM systems. Since audio systems operate with 16-24 bits and sampling frequencies of at least 44.1KHz, the necessary counter speed in this direct implementation is orders of magnitude higher than what can be realized in hardware. Even with innovative circuit design to reduce the effective counter speed [Ng85], a direct implementation is not practical without certain extensions. A more fundamental problem however is the nonlinearity within PCM-PWM conversion. The direct mapping of incoming digital PCM samples to a pulse width is a uniform sampling process (UPWM) that has significantly different characteristics than NPWM. Solutions to both the practical problems and inherent linearity problems within UPWM are the essential topics of the present chapter.

3.3

UPWM analysis

The topic of the following is a fundamental tonal analysis of uniformly sampled PWM. The analysis is an extension of the results that was presented on UPWM in [Ni97a]. Just as for NPWM, there are four fundamental variants of uniformly sampled PWM defined in Table. 3.1. The methods are illustrated in the time domain in Fig. 3.4 - Fig. 3.7. For coherence with the analysis methodology in chapter 2, the four variants of UPWM will be investigated in the following. The derivation of the DFS expressions is shown in detail in Appendix B.10. Only the resulting DFS expressions are given below DFS for UADS differential output J n (nπMq ) æ nπ ö sinç ny − nπq − ÷ n π q 2 ø è n =1 ∞

FUADS (t ) = − å +

1 − J 0 (mπM ) cos(mπ ) mπ m=1 ∞

å

(3.1)

J n (( nq + m )πM ) nπ sin(ny + mx − ) ( nq + m )π 2 m =1n = ±1 ∞ ±∞

−å

å

DFS for UBDS differential output J n (nπMq ) nπ cos(ny − nπq )sin( ) 2 nπq n =1 ∞

FUBDS (t ) = å

J (( nq + m )πM ) nπ +åå n cos(ny + mx ) sin( ) ( + ) 2 nq m π m =1n = ±1 ∞ ±∞

(3.2)

Chapter 3

71

1

1

2

3

4

5

6

7

8

A−side

1 0.5 0 0

1

2

3

4

5

6

7

8

1 0.5 0 0 1

1

2

3

4

5

6

7

8

−1 0

1

2

3

4

5

6

7

8

1 0.5 0 0

1

2

5

6

7

8

Comm.

Diff.

−1 0

B−side

0

0

3 4 Normalized time (t/tc)

Fig. 3.4 Time domain characteristics for UADS. f r = 81 .

1

1

2

3

4

5

6

7

8

1 0.5 0 0

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

−1 0

1

2

3

4

5

6

7

8

1 0.5 0 0

1

2

5

6

7

8

1 0.5 0 0 1

Comm.

Diff.

B−side

A−side

0 −1 0

0

3 4 Normalized time (t/tc)

Fig. 3.5 Time domain characteristics for UBDS. f r = 81 .

72

Digital Pulse Modulation

1

1

2

3

4

5

6

7

8

A−side

1 0.5 0 0

1

2

3

4

5

6

7

8

1 0.5 0 0 1

1

2

3

4

5

6

7

8

−1 0

1

2

3

4

5

6

7

8

1 0.5 0 0

1

2

5

6

7

8

Comm.

Diff.

−1 0

B−side

0

0

3 4 Normalized time (t/tc)

Fig. 3.6 Time domain characteristics for UADD. f r = 81 .

1

2

3

4

5

6

7

8

A−side

1

2

3

4

5

6

7

8

1 0.5 0 0 1

1

2

3

4

5

6

7

8

−1 0

1

2

3

4

5

6

7

8

1 0.5 0 0

1

2

5

6

7

8

Comm.

Diff.

1

1 0.5 0 0

B−side

0 −1 0

0

3 4 Normalized time (t/tc)

Fig. 3.7 Time domain characteristics for UBDD. f r = 18 . .

Chapter 3

73

DFS for UADD differential output M ö æ J n ç nπ q÷ nπ ö 2 ø æ è FUADD (t ) = å sinç ( q + 1) ÷ cos(ny ) 2 ø nπq è n =1 ∞

M æ J 0 ç mπ 2 +å è mπ m =1 ∞

ö ÷ ø cosæ mπ ö cos(mx ) ç ÷ è 2 ø

πM æ J n ç ( nq + m ) 2 è +åå π ( nq + m ) m =1n = ±1 ∞ ±∞

(3.3)

ö ÷ ø sinæ ( m + n(1 + q)) π ö cos(ny + mx ) ÷ ç 2ø è

DFS for UBDD differential output ∞

FUBDD (t ) = −4 å ∞ ±∞

−4å

å

m =1n = ±1

(

J n (nπq M2 )

n =1

nπq

J n ( nq + m ) π2M ( nq + m)π

sin(( q + 1)

nπ nπ nπ ö æ ) sin( ) sinç ny − ÷ 2 2 2 ø è

) sinæ (m + n(1 + q)) π ö sin( nπ ) sinæ ny + mx − nπ ö ç è

÷ 2ø

2

ç è

(3.4)

÷ 2 ø

3.3.1 UPWM harmonic distortion All components within the pulse-modulated output are summarized in Table 3.2. Compared to NPWM, uniformly sampling results in both phase and amplitude distortion of the fundamental. Furthermore, the output contains harmonics of the input leading to a finite total harmonic distortion. A parametric analysis has been performed on these important distortion characteristics of UPWM. Fig. 3.8- Fig. 3.11 shows THD calculated as an RMS sum of the first five harmonics vs. M and dBf r = 20 log( f r ) . The parameter space is chosen to represent worst-case conditions with maximal modulation and high frequencies. For all methods, THD is extremely dependent on frequency and modulation index. In the worst-case situation, none of the methods are sufficiently linear to honor the general linearity demands as e.g. THD < -80dB. For M < –20dB, all methods are sufficiently linear within the desired frequency range i.e. the linearity problems are exclusively present at high modulation index. Method

n'th harmonic of signal

UADS

J n (nπMq ) nπq

UBDS

J n (nπMq) nπ sin( ) nπq 2

UADD

M ö æ J n ç nπ q÷ è 2 ø nπ ö æ ÷ sinç (q + 1) è nπq 2ø

UBDD

M ö æ J n ç nπ q÷ è 2 ø nπ ö æ nπ ö æ ÷ sinç ÷ sinç (q + 1) è 2ø è 2ø nπq

m'th harmonic of carrier frequency 1 − J 0 ( mπM ) cos( mπ ) mπ

IM-component mx ± ny J n (( nq + m)πM ) ( nq + m)π

J n (( nq + m)πM ) nπ sin( ) 2 ( nq + m)π

Mö æ ÷ J 0 ç mπ è 2ø æ mπ ö÷ cosç è 2 ø mπ

πM ö æ ÷ J n ç ( nq + m) è 2 ø πö æ sinç ( m + n(1 + q )) ÷ è ( nq + m)π 2ø πM ö æ ÷ J n ç ( nq + m) è 2 ø πö æ πö æ sinç ( m + n(1 + q)) ÷ sinç n ÷ è ( nq + m)π 2ø è 2ø

Table 3.2 Summary of the components that constitute the UPWM DFS expressions.

74

Digital Pulse Modulation

0

−2

−5

0

−4

−6

0 −4

Modulation index(dB)

−6

0

−7

0

−8

−5

0

−6 0

−10

−12

−7

−8

0

0

−14

−6

0

−9

0

−16

−18

−55

−50

−45

−40 −35 Frequency (dBfr)

−30

−25

−20

Fig. 3.8 Contour plot of THD vs. modulation index and frequency ratio dBf r for UADS. Level curves of constant THD draw a straight line in the (M,fr)- parameter space.

0

10

−1 −2

0

−7

−8 0

0

00

−9

−1

−4

0

−5

−7 0

−1

0

−8

0 −9

00

−1

30

20

−1

−10

0

10

−6

−1

−8

0

14

Modulation index(dB)

−6

−12

10

−1

−14

0

0 −9

0

0 −1

30

20

−8

−1

−1

−16

40

−1

−18

−55

−50

−45

−40 −35 Frequency (dBfr)

−30

−25

−20

Fig. 3.9 Contour plot of THD vs. modulation index and frequency ratio dBf r for UBDS.

Chapter 3

75

0

−2

−4

−9 0 0 −1

−8

0 0

0

20

−9 0

30

−12

−7

1 −1

−1

−10

−1

Modulation index(dB)

0 −5

0 −8

0 −6

−6

0 −6

0 −8

−14

−1 00 0

20

10

−7

−1

−1

−16

−18

0

−9

−55

−50

−45

−40 −35 Frequency (dBfr)

−30

−25

−20

Fig. 3.10 Contour plot of THD vs. modulation index and frequency ratio dBf r for UADD.

0

−2

−8

0

0

20

−1

00

−9

−1

−4

0

−7

−6

30 −9

−8

0

20 −1

40

00 −1

−1

−10

0

−12

10

−1

50

−1

−1

Modulation index(dB)

10

−1

−1

−8

30

−14

6 −1 0 −1

−1 −40 −35 Frequency (dBfr)

20

40 −45

00

−1

−16

−18

10

−1

−55

−50

−30

−25

−20

Fig. 3.11 Contour plot of THD vs. modulation index and frequency ratio dBf r for UBDD.

76

Digital Pulse Modulation

The distortion characteristics of UPWM differ significantly between methods. The maximal frequency ratios corresponding for below –60dB and –80dB THD are:

f r ,max (THD