Chapter 5



Pulse modulation








Mind map：  





Mind map for 
Chapter 5








In Chapter 4, some parameters of a sinusoidal carrier wave are varied continuously in accordance with the message signal. In this chapter, some parameters of a pulse train are varied in accordance with the message signal. The carrier wave is a pulse, which is different from CW modulation.

The following issues will be discussed：  

 Sampling, which is basic to all forms of pulse modulation.

 Pulse amplitude modulation (PAM，脉冲幅度调制) (In discrete amplitude form).

 Quantization (量化), after the process of sampling,in discrete form in both amplitude and time.

 Pulsecode modulation(PCM，脉冲编码调制), which is the standard method for the transmission of an analog message signal by digital means.

 TDM (时分复用).

 Delta modulation(DM/ΔM，Δ调制/增量调制).

Differential pulsecode modulation(DPCM，差分脉冲编码调制).





视频讲解


5.1Sampling process

The process of digitizing an analog signal is illustrated in Figure 5.1.1. There are three main steps in the process of digitizing an analog signal：   sampling, quantization and encoding.



Figure 5.1.1The process of digitizing an analog signal



1. The sampling theorem (the sampling of lowpass analog signals)

If the highest frequency of a continuous analog signal s(t) is less than fH, and if it is sampled by periodic impulses with interval time T≤1/2fH, then s(t) can be completely decided by these samples,as shown in Figure 5.1.2. 




Figure 5.1.2Sampling process


The ideal form of sampling process is called instantaneous sampling (瞬时采样).Ts is the sampling period, and fs=1Ts  is the sampling rate.

The derived process is in the following Table 5.1.1：   let the spectra of m(t), δT(t) and  ms(t) be expressed by M(f),ΔΩ(f) and Ms(f). If the sampling frequency is lower than 2fH,the adjacent spectra will be superposed(重叠), hence the original signal spectrum M(f) could not be separated correctly by LPF at the receiver.


Table 5.1.1The derivation of sampling process



m(t)
M(f)
δT(t)=δ(t-nTs)ΔΩ(f)=1T∑∞n=-∞δ(f-nfs)

ms(t)=m(t)·δT(t)Ms(f)=M(f)*ΔΩ(f)=1T∑∞-∞M(f-nfs)


From Figure 5.1.2, the condition of restoration of original signal is fs≥2fH.The lowest sampling frequency  2fH is called Nyquist sampling rate(奈奎斯特采样速率).The corresponding largest sampling time interval is called Nyquist sampling interval(奈奎斯特采样间隔).

2. The sampling of bandpass analog signals 

The frequency band of bandpass signals is limited between fL and fH.B=fH-fL is the bandwidth of the analog signal. Here, we only give the result. The sampling frequency fs can be written as：   


fs=2B+2KBn=2B1+Kn
(5.1.1)


where n is the largest integer (整数) less than fH/B,0<k<1.The relation between fs and fL is shown in Figure 5.1.3.




Figure 5.1.3The relation between fs and fL



When fL=0,fH=2B, this is the sampling condition for the low pass filter.

When fL0,fs→2B, the signal is a narrowband signal.

5.2Analog pulse modulation (模拟脉冲调制)

There are three parameters of pulse：   amplitude, width (duration) and position. Therefore, we can get three different pulse modulations：   PAM (pulse amplitude modulation), PDM (pulse duration modulation) and PPM (pulse position modulation), as shown in Figure 5.2.1.

Although these modulations are discrete in time, they are still analog modulations. In order to convert an analog signal to a digital signal, quantizer should be used in the next step.



Figure 5.2.1Three types pulse modulation



In this part, the PAM is introduced in detail. There are two types PAM：   natural sampling (自然采样) and flattop sampling (平顶采样).

1. Natural sampling PAM

As can be seen from Figure 5.2.2, the top of each pulse is natural.



Figure 5.2.2Natural sampling PAM



The differences between sampling and PAM is the period signal s(t), which is the period pulses.

According to


ms(t)=m(t)·s(t)


Its corresponding Fourier transform is


Ms(f)=M(f)*S(f)
=AτT∑∞n=-∞sinc(πnτfH)M(f-2fH)



Because the carrier wave is periodic rectangular pulses, the envelope of Ms(f)
 is the sinc function.The original signal M(f) can be recovered/reconstructed when Ms(f) passes through a LPF at the receiver (see Figure 5.2.3).



Figure 5.2.3The waveform and spectrum of PAM



2. Flattop sampling PAM

The top of PAM is flat, as shown in Figure 5.2.4.



Figure 5.2.4Flattop sampling PAM



There are two operations involved in the generation of the PAM signal, as shown in Figure 5.2.5.

(1) Instantaneous sampling (瞬时采样).

The sampling rate fs is choosed in accordance with the sampling theorem.

(2) Lengthening (保持) the duration of each sample so obtained to some constant value T.



Figure 5.2.5The principle of PAM



Where


h(t)=1,0<t<T
12,t=0,t=T
0,otherwise

H(f)=Tsinc(fT)exp(-jπfT)


In Table 5.2.1, the comparisons between time and frequency domain equations are given.


Table 5.2.1The derivation of PAM



Time domain
Frequency domain
ms(t)=m(t)·δT(t)=∑∞n=-∞m(nTs)·δ(t-nTs)
Ms(f)=M(f)*Δn(f)=fs∑∞n=-∞M(f-nfs)
mH(t)=ms(t)*h(t)MH(f)=Ms(f)·H(f)=fs∑∞n=-∞
M(f-nfs)·H(f)



The problem is how to recover the original message signal m(t).The first step is that mH(t) passes through a LPF（low pass filter）. The spectrum of LPF output is equal to M(f)H(f). Obviously, there is frequency distortion because of H(f). This distortion may be corrected by connecting an equalizer in cascade (串联) with the LPF, as shown in Figure 5.2.6.



Figure 5.2.6The reconstruction of message signal



The magnitude response of the equalizer is given by


1|H(f)|=1Tsinc(fT)=πfsin(πfT)



For a duty cycle(占空比)T/Ts≤0.1, the amplitude distortion is less than 0.5%, in 
which case the need for equalization may be omitted altogether.





视频讲解


5.3Quantization process (量化过程)of sampled signal

Amplitude quantization：   the process of transforming the sample amplitude m(kT) of a message 
signal m(t) at time t=kT into a discrete amplitude mq(kT) taken from a finite set of possible 
amplitudes(有限个可能的取值).An example of quantization process is shown in Figure 5.3.1. 



Figure 5.3.1The quantization process



m(kT) expresses the sampled signal input to a quantizer, mq(kT)expresses the quantized value of the output signal of this quantizer, as shown in Figure 5.3.2.



Figure 5.3.2Quantizer



q1~q6 are 6 possible levels of the quantized signal. 

m1~m5 are the boundary points of the quantization intervals.


mq(kT)=qi,mi-1≤m(kT)≤mi
(5.3.1)


In this example, M intervals are with equal space, so it is called uniform quantization (均匀量化). If the M intervals are with unequal spaces, it is called nonuniform quantization (非均匀量化), which is based on the uniform quantization.

nonuniform quantizationuniform quantization

5.3.1Uniform quantization (均匀量化)

Assume the magnitudes of the sampled signal is within a~b, and the number of quantization levels is M, so the quantization interval (量化间隔) is Δν=b-aM (also called stepsize：  步长), and the boundary points of the quantization intervals are mi=a+iΔν,i=0,1,…,M.

Obviously, quantized output mq is different from signal sample before quantization mk, there is an error of quantizer. This error is usually called quantization noise (量化噪声) and signal power to quantization noise power ratio (信噪比) is used for estimating the influence of the error to the signal.

The average value Nq of the quantized noise power for uniform quantization can be expressed by 


Nq=［(mk-mq)2］=∫ba(mk-mq)2f(mk)dmk
=∑Mi=1
∫mimi-1(mk-qi)2f(mk)dmk
 (5.3.2)


where f(mk) is the probability density of signal sample mk


mi=a+iΔν
qi=a+iΔν-Δν2
(5.3.3)



The average power of signal mk can be expressed as 


S=E(m2k)=∫bam2kf(mk)dmk
(5.3.4)


Example 5.3.1: Assume the number of quantization levels for a uniform quantizer is M, and the sample of the input signal has uniform probability density in the interval［-a,a］. Find the average signal to quantization noise ratio for the quantizer.

Solution：  From equation(5.3.2), we can get


Nq=∑Mi=1∫mimi-1(mk-qi)2f(mk)dmk=∑Mi=1∫mimi-1(mk-qi)212admk
=∑Mi=1∫-a+iΔν-a+(i-1)Δνmk+a-iΔν+Δν22·12admk
=∑Mi=112a·Δν312=M(Δν)324a


According to M·Δν=2a,  so the quantization noise power is 


Nq=(Δν)212



The average power of signal is


S0=∫a-am2k12admk=M212·(Δν)2



The the average signal to quantization noise ratio is：  

S0Nq=M2or
S0NqdB=20lgM(dB)



S0Nq is increasing along with the increasing of number M. That is to say when the signal is small, signal to quantization noise ratio is also small. For the speech signal, there are lots of small components. To overcome this disadvantage, the nonuniform quantization is often used in practical applications.

5.3.2Nonuniform quantization

A nonuniform quantizer is equivalent to passing the sampled signal through a compressor and then applying the compressed signal to a uniform quantizer, as illustrated in Figure 5.3.3.



Figure 5.3.3The equivalent process of nonuniform quantizer


Compression uses a nonlinear circuit to convert input voltage x to output voltage y：  

y=f(x)



where x has nonuniform scale and y has uniform scale.

Theoretically,f(x) should be a logarithm(对数)function.f(x) needs to be properly modified in different requirements of the practical condition.There are two ITU recommended logarithm compression laws (对数压缩定律)：   Alaw and μlaw, for telephone signal. The logarithm laws are give in Table 5.3.1.


Table 5.3.1Logarithm compression laws



Alawμlaw
y=Ax1+lnA,0<x≤1A
1+lnAx1+lnA,1A≤x≤1
y=ln(1+μx)ln(1+μ)

13 segments
15 segments
E1：   China & European countries
T1：   North American, Japan and Korea


Alaw 13 segments method：   Alaw compression is a continuous smooth curve. It is difficult to be accurately achieved by electronic circuit, while the 13 segments method can be easily achieved by digital circuit approximately, which is plotted in Figure 5.3.4.



Figure 5.3.4Alaw 13 segments


In Table 5.3.2, the slope of each segment is given. As can be seen, the first and second segments have the same slope.


Table 5.3.2Slope of each segment



No.(折线段号)
1
2
3
4
5
6
7
8
slope(斜率)
16
16
8
4
2
1
1/2
1/4


Why we called it 13 segments?

The input voltage of speech signal has positive and negative polarities. So the above compression curve is only one half of the practical characteristic curve of the compressor. There is another half in the 3rd quadrant (象限), so the curve is odd symmetry to the origin (关于原点奇对称).

The slopes of the 1st and 2nd segments in 1st quadrant and 3rd quadrant are the same. These 4 segments compose a straight line. Therefore, it is called 13 segments.

Signal after quantization is a digital signal with discrete values. The next step is how to encode these discrete values.





视频讲解


5.4PCM(pulse code modulation,脉冲编码调制)

PCM is the most basic form of digital pulse modulation.

PCM：   a message signal is represented by a sequence of coded pulses which is accomplished by representing the signal in discrete form in both time and amplitude(PCM编码后的信号在时间和幅度上的取值都是离散的).

The most often used code is to use the binary symbols representing discrete values, such as 0 and 1.The basic operations performed in the transmitter of a PCM system are sampling, quantizing and encoding, as illustrated in Figure 5.4.1. The quantizing and encoding operations are usually performed in the same circuit, which is called an analogtodigital converter(A/D转换器).The basic operations in the receiver is the regeneration of the message signals：   decoding and reconstruction.



Figure 5.4.1The PCM process



5.4.1The principle of PCM

There are 8 bits in PCM code word(码字)，which can satisfy the speech communication quality.

c1—the polarity bit(极性位) +,c1=1-,c1=0； 

c2c3c4—segment bits(段落码)：  000~111；  

c5c6c7c8—innersegment bits (段内码)：   000~1111(16 innersegments in each segment).


Innersegment bits are uniformly encoded according to quantization intervals, but quantization intervals of different segments are different.

Among these 8 segments, the lengths of 1st and 2nd segments are the shortest 1/128, and slopes of them are the largest, after it is equally divided into 16 subsegments, the each subsegment equals：  


Δ=1128×116=12048



which is called the smallest quantization interval(最小量化间隔).

If uniform quantization is used to keep the same dynamic range 1/2048, then 11 bits code word is necessary.

While for the PCM systems, only 7 bits (不含极性位) are used, which embody the advantage of nonuniform quantization.

The typical sampling rate of telephone signal is 8kHz (≥2×3.4kHz).Thus the typical transmission bit rate of digital telephone is 64kb/s in PCM system. This rate has been adopted by ITUT recommendation.


Example 5.4.1：   Using Alaw 13 segments encoding, if the sampling value is +1270, find the output of PCM encoder and the quantization error.

Answer：    (1) First, since +1270>0, the polarity bit c1=1；  

(2) Second, 1270 is in the 8th segment, so c2c3c4=111；  

(3) Finally, determine which innersegment 1270 is in. There are usually two methods.
One is the successive comparison (逐次比较). Another one is direct calculation. The 8th segment is plotted in Figure 5.4.2.



Figure 5.4.2The quantization interval of 8th segment


Successive comparisons：  


1270<1536,c5=0
1270<1280，c6=0
1270>1152，c7=1
1270>1216，c8=1


Direct calculation：  

innersegment quantization interval is (2048-1024)/16=64；  

Since (1270-1024)/64≈3.8, the inner segment coding is c5c6c7c8=0011.

Quantization error is：   1270-(1216+1280)/2=22Δ.

Therefore, the output of PCM encoder is 11110011 and the quantization error is 22Δ.

5.4.2Noise in PCM system

The derivation signal to quantization noise ratio of the uniform quantizer has been discussed in Section 5.3.1.


S/Nq=M2



When N bits binary code word is used for encoding, the above equation can be written as 

S/Nq=22N(M=2N)



This equation shows that S/Nq of PCM system is only related to N and increases with N exponentially. For a lowpass signal, the sampling rate should be no less than 2fH, so in the PCM system, this is equivalent to not less than  2NfHb/s. The system bandwidth is at least B=NfH.


S/Nq=22(B/fH)



The S/Nq of PCM system increases with the bandwidth B exponentially.

5.4.3Delta modulation (增量调制)
1. Principle

Delta modulation(DM or ΔM)provides a staircase approximation to the oversampled (i.e.,at a rate much higher than the Nyquist rate) version of the message signal, as illustrated in Figure 5.4.3.



Figure 5.4.3The waveform of DM


The difference between the input and the approximation is quantized into only two levels, namely,±Δ, corresponding to positive and negative differences.

If the approximation falls below the signal at any sampling epoch, it is increased by Δ.If the approximation lies above the signal, it is diminished by Δ. 

The principal (主要的)virtue of delta modulation is its simplicity. It may be generated by applying the sampled version of the incoming message signal to a modulator that involves a comparator (比较器), quantizer and accumulator interconnected as shown in Figure 5.4.4. The block labeled z-1 inside the accumulator represents a unit delay, that is, a delay equal to one sampling period.



Figure 5.4.4DM system


The basic principle of Δ modulation is in the following：  


e［n］=m［n］-mq［n-1］
eq=Δsgn［e(n)］
mq［n］=mq［n-1］+eq［n］



The quantizer output mq［n］ is coded to produce the DM signal. The rate of information transmission is simply equal to the sampling rate fs=1Ts. 

2. Quantization error

There are two types of quantization error：   slope overload distortion and granular noise (see Figure 5.4.5).



Figure 5.4.5Two different forms of quantization error



If considered the maximum slope of the original input waveform m(t), it is clear that in order for the sequence of samples {mq［n］} to increase as fast as the input sequence of samples {m［n］} in a region of maximum slope of m(t), we require that the following condition：  


ΔTs≥maxdm(t)dt



be satisfied. Otherwise, we find the stepsize Δ is too small for the staircase approximation mq(t). In contrast to slopeoverloaded distortion, granular noise is analogous (类似) to quantization noise in a PCM system.

5.4.4DPCM(differential pulse code modulation，差分脉冲调制)

When a voice or video signal exhibits a high degree of correlation between adjacent samples (邻近的采样) the resulting PCM encoded signal will contain redundant information (冗余信息).

In order to remove this redundancy before encoding, we use a more efficient coded signal, which is the basic idea of DPCM, as shown in Figure 5.4.6.



Figure 5.4.6DPCM system



The input signal of the DPCM quantizer is 


e［n］=m［n］-m^［n］



where m［n］ is the unquantized sample signal and m^［n］ is the prediction of m［n］.

The relationship between the output and input of the predictor is 


m^=∑qi=1aimq(线性预测) 


where p is the prediction order and ai is the prediction coefficient(系数).The predicted value is the weighted sum of previous p samples of the signal with quantization error.

If p=1,a1=1 then m^=mq-1.The predictor is simply a delay circuit, and the delay time is sampling interval T. If the quantizer is onebit (twolevel),the DPCM is ΔM,i.e.ΔM is the special case of DPCM.

5.5TDM(timedivision multiplexing,时分复用)

The concept of TDM is shown in Figure 5.5.1. Each input message signal is first restricted in a lowpass antialiasing filter. The functions of each part are described below. 



Figure 5.5.1Block diagram of TDM system


LPF：   restricts each input message in bandwidth and removes the nonessential frequencies.

Commutator (换向器)： which is usually implemented using electronic switching circuitry. The function is to sequentially interleave these N samples inside. The sampling interval is T. It is the essence of TDM operation.

Pulse modulator：   transform the multiplexed signal into a form suitable for transmission over the common channel.

Pulse demodulator：   the reverse operation of pulse modulator.

Decommutator：   operates in synchronism (同步) with commutator in the transmitter.

The multiplexing of digital signals is accomplished by using a bitbybit interleaving procedure with a selector switch that sequentially takes a bit from each incoming line and then applies it to the highspeed common line. At the receiving end of the system the output of this common line is separated out into its lowspeed individual components and then delivered to their respective destination, as plotted in Figure 5.5.2.



Figure 5.5.2Principle of TDM


Summary and discussion

In this chapter, we introduced the process of analog signal digitization：   sampling, quantization and encoding.

(1) Sampling, which operates in the time domain；   the sampling process is the link between an analog waveform and its discretetime representation, which also is an analog signal in amplitude.

(2) Quantization, which operates in the amplitude domain；   the quantization process is the link between an analog waveform and its discreteamplitude representation.

(3) Encoding, which operates in both time and amplitude domain, encoding process is the link between discreteamplitude and binary representation.

The sampling process is based on the sampling theorem, which states that when an analog signal with frequency band within (0, fH) is sampled, the lowest sampling rate should not be less than Nyquist sampling rate 2fH. The sampled signal may be converted to different analog pulse modulation signals, including PAM, PDM and PPM. In TDM of several channels, signal processing usually begins with PAM. In order to make full use of the time interval of each sampling point, TDM is designed and applied in speech communication system.

There are two methods for the quantization of a sampled signal, one is uniform quantization, another one is nonuniform quantization. Nonuniform quantization is usually applied in speech signal with the logarithm characteristic recommended by ITUT, i.e., Alaw and μlaw, which may effectively improve signal to quantization noise ratio, especially the small amplitude signal. European countries and China adopt Alaw；   North American countries and Japan as well as other counties and areas adopt μlaw. 13 segment and 15 segment methods are applied in digital circuits to achieve the Alaw and μlaw quantization.

Signal after quantization is already a digital signal. Encoding methods, such as PCM, DPCM and ΔM, are usually used to convert a quantized signal into a binary signal. This process is lossy in the sense that some information is lost, but the loss of information is under the designer’s control in that it can be made small enough.

The signal to quantization ratio of PCM system increases with the bandwidth B exponentially, while the analog modulation increases linearly. That is why PCM and its improved coding are widely used. 

Homework 

5.1Suppose the spectrum of a message signal m(t) is M(f),its expression is 


M(f)=1-|f|200,|f|<200Hz
0,otherwise




(1) If the sampling rate is 300Hz, try to draw the spectrum of the sampled signal ms(t)；  

(2)  If the sampling rate is 400Hz, try to redraw it.

5.2Using Alaw 13 segment encoding, if the sampling value is 635, find the output of PCM encoder.

5.3Suppose the message signal is m(t)=9+Acosωt, where A≤10V. If m(t) is quantized to 40 levels, try to find the number of binary bits and the quantization intervals.


Terminologies


sampling采样

quantizing量化

encoding编码

uniform quantization均匀量化

nonuniform quantization非均匀量化

quantization noise量化噪声

equalizer均衡器

compressor压缩器

commutator换向器

PCM脉冲编码调制

DPCM差分脉冲编码调制

ΔM增量调制

PAM脉冲幅度调制

PDM脉冲宽度调制

PPM脉冲位置调制

TDM时分复用