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Chapter 2 Discrete Pulse Modulation 2.1

Introduction

Information could be transmitted in analog, discrete or digital form. Discrete signal transmission was implemented long time ago before digital signal were experienced. The discretization of the analog signal led to the digital communication approaches later. In this chapter, generating the discrete and binary signals are introduced by introducing sampling theory.

2.2

Sampling Theory

In order to transmit the analog information using digital system, it has to be converted into a digital form. Analog-to-digital conversion (ADC) is the most popular process that is used in almost all digital processing. The ADC samples the analog signal, hold its value for short period of time, and then quantize the sampled value to certain discrete voltage levels. This process is shortened in the term sample and hold (S/H), and a basic electrical circuit that can implement this process is shown in Figure 2.1.

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Ts

Analog signal g(t)

C Sampled signal gδ (t)

Figure 2.1: Sample and hold block diagram

Figure 2.2: Sampling g(t) in time domain.

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The samples are taken for each Ts span or interval of time, where Ts = 1/fs and fs is the sampling rate. Suppose the energy signal x(t) has a bandwidth of B Hz, then its sampled version in time domain xδ (t) is: xδ (t) =

1 X

n=1

x(t) (t − nTs )

(2.1)

where T s = 1/fs is the sampling interval, i.e. the time you wait after taking a sample before you pick the next one. Note that equation 2.1 is simply the sum of the analog signal multiplied by an impulse train (t), where the impulses are separated by the sampling interval Ts . It is important to understand the spectrum of the sampled signal in the frequency domain, because it will facilitate the communication system design. Therefore, Fourier transform is used to obtain Xδ (f ), as the signal in time domain could be expressed in frequency domain using the following Fourier transform relationship: FT

xδ (t) ←−−−−→ Xδ (f ) Hence:

1 X ⇤ ⇥ (t − nTs ) Xδ (f ) = FT xδ (t)

(2.2)

(2.3)

n=1

which is simply:

⇥ ⇤ FT Product of x(t) and impulse train

(2.4)

One of the Fourier transform properties stated that the multiplication in time domain is a convolution of the spectrum of each signal in the frequency domain. Therefore, the relationship in formula 2.4 will be: X(f ) = X(f ) ∗ F T {(t − nTs )} 1 n 1 X (f − ) = X(f ) ∗ Ts Ts n=1 = X(f ) ∗ fs

1 X

n=1

(f − nfs )

(2.5) (2.6) (2.7)

Using the linearity of convolution, the last equation will be:

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Figure 2.3: The spectrum of the analog signal compared to the spectrum of the sampled version of it. Here A is the amplitude, 1T s is the sampling frequency fs and 2W is the signal bandwidth.

1 X

Xδ (f ) = fs

n=1

X(f ) ∗ (f −

n ) Ts

(2.8)

The right-hand-side (RHS) of the final equation could be interpreted as follows: Xδ (f ) consists of replicas of the analog signal spectrum X(f ), where each one of them is shifted by sampling frequency fs as shown in 2.3. Mathematically, the spectrum of the sampled signal is: Xδ (f ) = fs

1 X

n=1

2.3

X(f − nfs )

(2.9)

Nyquest Theorem and Aliasing

In order to retrieve the original analog signal from its samples, the sample rate fs has to be equal or higher than the maximum frequency of it. This rule is commonly known as Nyquest theorem. A band limited signal of finite energy which has no frequency higher than fmax Hertz may be completely recovered into its analog form if: fs > 2fmax

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(2.10)

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Figure 2.4: (a) Spectrum of a signal. (b) Spectrum of an undersampled version of the signal, exhibiting the aliasing phenomenon. The digital to analog conversion (DAC) is required for some applications like speech communication. If the condition in equation 2.10 is met, then the DAC will produce the analog signal exactly as it was before sampling. On the other hand, if the analog signal is sampled with a rate that is lower than fmax , then the DAC will retrieve a dif ferent signal. In this case, the DAC output is called alias of the original signal. The aliasing problem could be solved by making sure that the analog signal is sampled in rate that is higher than it maximum frequency. In many cases, the analog signal might have undesired frequency components that are higher than the designer thought. If removing those unexpected frequency components do not affect the message that will transmitted, then they should be filtered out using what is known as Anti-Aliasing Filter (AAF). In most cases, AAF is a LPF; as most of the messages are low-pass signals. However, the AAF could be BPF in some cases, like when the ADC is required to deal with specific tone in music application. Example 2.1 Determine the sampling rate and interval of the signal: x(t) = 2cos(4000⇡t)cos(1000⇡t) .

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Figure 2.5: Spectrum of the message signal. Solution: x(t) = 2cos(4000⇡t)cos(1000⇡t) 1⇥ ⇤ =2 cos((4000 − 1000)⇡t) + cos((4000 + 1000)⇡t) 2 = cos(3000⇡t) + cos(5000⇡t) = cos(2⇡f1 ) + cos(2⇡f2 ) Comparing the frequencies, f1 = 1500 and f2 = 2500. Hence, the highest frequency of the signal x(t) is: fmax = f2 = 2500 Hence, sampling rate is: fs = 2fmax = 2f2 = 5000

Example 2.2 Figure 2.5 shows the spectrum of a message signal, which was sampled by 1.5fmax , where fmax = 1 Hz is the maximum signal frequency. 1. Sketch the spectrum of the sampled version of the signal. 2. When the sampled signal is received, it has to go through a low-pass filter (LPF). If the cut-of f frequency of the LPF is 1 Hz, sketch the spectrum of the output signal from this filter. Solution: 1. When x(t) is sampled, its spectrum will be: CHAPTER 2. DISCRETE PULSE MODULATION

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Figure 2.6: (a) Spectrum of sampled signal. (b) Transfer function of LPF. (c) Filtered signal.

Xδ (f ) = fs

1 X

X(f − nfs )

1 X

X(f − 1.5nfmax )

1 X

X(f − 1.5n)

n=1

Here fs = 1.5 ;max , hence it will be: Xδ (f ) = 1.5fmax

n=1

With Fmax = 1 Hz, the above equation will be: Xδ (f ) = 1.5

n=1

Figure 2.6(a) is the plot of the above equation. 2. As the signal is sampled with fs = 1.5fmax an aliasing effect occurs as shown in 2.6(a). When the sampled signal is passed through a LPF of BW = fmax [see figure 2.6(b)], the output will be as shown in 2.6(c). Note that the signal in 2.5 and 2.6(c) are not the same because of the aliasing effect phenomenon.

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Figure 2.7: Pulse Amplitude Modulation (PAM).

2.4

Pulse Amplitude Modulation

In pulse-amplitude modulation (PAM), the amplitudes of regularly spaced pulses are varied in proportion to the corresponding sample values of a continuous message signal; the pulses can be of a rectangular form or some other appropriate shape. Pulse-amplitude modulation as defined here is somewhat similar to natural sampling, where the message signal is multiplied by a periodic train of rectangular pulses. However, in natural sampling, the top of each modulated pulse is allowed to vary with the message signal, whereas in PAM it is retained flat. The waveform of a PAM signal is illustrated in Figure 2.7. The dashed curve in this figure depicts the waveform of the message signal m(t), and the sequence of amplitude-modulated rectangular pulses shown as solid lines represents the corresponding PAM signal s(t). There are two operations involved in the generation of the PAM signal: 1. Sampling the signal of the message m(t) every sTs seconds, where the sampling rate fs = 1/T s is selected based on sampling theorem. 2. Increasing the duration length of each sample, so that it occupies some finite value T . Those operations are the same function of the sample-and-hold (S/H) circuit discussed earlier in section 2.2. The reason for increasing the duration length of a sample is to minimize the required channel bandwidth (BW), as BW is inversely proportional to pulse duration T .

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If the sequence of pulses in PAM in figure 2.7 is denoted as s(t), then PAM signal could be described as: s(t) =

1 X

n=1

m(nTs )h(t − nTs )

(2.11)

where Ts is the sampling interval, m(nTs ) is the sampled version of m(t) at time t = nTs and h(t) is the standard pulse train. The latter is has a rectangular shape, duration T and repeated at each sampling interval (this is why it has the argument t − nTs ). Therefore: 8 > ✓ < 1, 0 < t < T T ◆ t− 2 1 (2.12) = h(t) = rect , t = 0, t = T 2 > T : 0, otherwise PAM Receiver At the receiver, the PAM is sampled using the same sampling rate fs = T1 , to make mδ (t) which is: s mδ (t) =

1 X

n=1

m(nTs ) (t − nTs )

(2.13)

where (t − nTs ) is the time-shifted delta function (comb function). Convolve mδ (t) with pulse in equation 2.12 to get: Z 1 mδ (t) ∗ h(t) = mδ (⌧ )h(t − ⌧ )d⌧ 1 1

= =

Z

1 X

1 n=1 Z 1 1

X

1 n=1 1

=

X

m(nTs ) (t − nTs )h(t − ⌧ )d⌧ (2.14) m(nTs ) (t − nTs )h(t − ⌧ )d⌧

m(nTs )

n=1

Z

1 1

(t − nTs )h(t − ⌧ )d⌧

As delta function has only values at t − nTs , multiplying it by h(t − ⌧ ) produces h(t − nTs ). Therefore the convolution will be: mδ (t) ∗ h(t) =

CHAPTER 2. DISCRETE PULSE MODULATION

1 X

n=1

m(nTs )h(t − nTs )

(2.15)

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which equals the PAM transmitted signal in equation 2.11. In words, receiver can detect PAM if it samples the received signal and convolve it with the carrier rectangular pulses, that have the same frequency at the transmitter. NOTE: Multiplying the sampled received signal by the carrier is a common technique in demodulation. PAM Receiver Design In fact, PAM signal s(t) could be obtained at the receiver by convolving the received signal with the rectangular pulse train as in equation 2.15, which is: s(t) = mδ (t) ∗ h(t) =

1 X

n=1

m(nTs )h(t − nTs )

(2.16)

Now, take Fourier transform of the above equation, to convert convolution to multiplication as the latter could be easily implemented using electronic devices, which will lead to: S(f ) = Mδ (f )H(f )

(2.17)

where Mδ (f ) is the Fourier transform of the sampled received PAM, which is related to the continuous PAM as follows (using sampling theory 2.9): Mδ (f ) = fs

1 X

k=1

M (f − kfs )

(2.18)

where fs is the sampling rate. Therefore: S(f ) = fs

1 X

k=1

M (f − kfs )H (f )

(2.19)

In order to PAM signal back to its analog origin, all extra parts in the previous equation should be removed to get pure S(f ). So, electronic devices are used in PAM receiver to do the following: 1. Reconstruction: Pass the signal in 2.19 through a reconstruction filter, which an LPF which has a characteristics shown in figure 2.8. 2. Equalization: Because flat-top sampling is the aspect of PAM, amplitude distortion and a delay of T /2 is expected at the receiver (aperture effect ). Hence, the reconstructed signal is step 1 is passed through an equalizer, which has to remove the flat-top sampling effect CHAPTER 2. DISCRETE PULSE MODULATION

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Figure 2.8: Reconstruction filter ideal amplitude response

Figure 2.9: PAM receiver: A system that recovers message m(t) from PAM signal. by having a magnitude response that is reciprocal of the rectangular samples H(f ). The PAM equalizer response is: 1 1 2⇡ = = sin(2⇡f ) H(f ) T sinc(f T )

(2.20)

Using those logical steps, received PAM signal can be back to the original message m(t). The resultant PAM receiver could be as shown in 2.9.

2.5

Pulse Width Modulation

In PAM, the amplitude of the pulse is the variable parameter that represents the amplitude of the message. The width (duration) of the pulse could also be used to indicate the message voltage level for modulation. In pulse-width modulation (PWM), the message signal samples are used to vary the width of the pulse train signal. This form of modulation is also referred to as pulse-duration modulation (PDM) or pulse-length modulation (PLM). The modulating message signal can vary the time of occurrence of the rising edge, the falling edge, or both edges of the pulse. Figure 2.10 depicts an example of PWM where the modulating signal is sinusoidal. In PWM, the carrier pulse train has a variable duration so that the mod-

CHAPTER 2. DISCRETE PULSE MODULATION

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Figure 2.10: Pulse-width modulation (a) Modulating message signal (b) Carrier pulses (c) Modulated PWM signal.

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ulated signal s(t) will be : s(t) =

1 X

n=1

˙ − nTs ) h(t

˙ is the variable-width PWM signal, which is: where h ◆ ✓ t − T2 ˙h = rect wherea ∝ m(t) aT

(2.21)

(2.22)

where m(t) is the modulating signal (message). Therefore, the sampling rate relies on the maximum expected input voltage, of the message signal, and has the following relationship with it: Ts ∝

1 amax T

(2.23)

The maximum expected voltage in the message signal will degrade the sampling rate. If the expected maximum voltage increases, the PWM signal width increases as well, and hence the transmitter power will also be increased, which is the main advantage of the PWM. The PWM receiver is simply a counter and voltage comparator. The counter counts the clock pulses when the PWM is high and decode the message as a proportion of those pulses. Low number of pulses on high PWM signal means low voltage and vice versa.

2.6

Time Division Multiplexing

The sampling theorem provides the basis for transmitting the information contained in a band-limited message signal m(t) as a sequence of samples of taken uniformly at a rate that is usually slightly higher than the Nyquist rate. An important feature of the sampling process is a conservation of time. That is, the transmission of the message samples engages the communication channel for only a fraction of the sampling interval on a periodic basis, and in this way some of the time interval between adjacent samples is cleared for use by other independent message sources on a time-shared basis. We thereby obtain a time-division multiplex (TDM) system, which enables the joint utilization of a common communication channel by a plurality of independent message sources without mutual interference among them. Suppose a TDM communication system shown in figure 2.11 has N messages to be transmitted. According to Nyquest theorem, the commutator CHAPTER 2. DISCRETE PULSE MODULATION

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Figure 2.11: Time division multiplexing (TDM) system. (or sampling switch) has to take a sample from each message at a rate offs . The receiver will use a the same rate to decode the time division multiplexed messages and separate them eventually. If the highest frequency among those messages is W , then the sampling frequency fs of the commutator is: fs ≥ 2W

(2.24)

Hence the time space between successive samples of any message (signal) is: 1 fs 1 ) Ts ≤ 2W Ts =

(2.25) (2.26)

The sample time interval Ts contains one sample from each input message, and each time interval is called a frame. For N messages, in each frame there will be one sample of the N messages, which means one frame of interval Ts will have N samples. Therefore: Spacing between two samples =

Ts N

(2.27)

The signaling rate, which is the number of samples per second at the TDM output, channel and the receiver input, will be: 1 Spacing between two samples 1 = Ts /N N = Ts = N fs

TDM Signaling Rate =

CHAPTER 2. DISCRETE PULSE MODULATION

(2.28)

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As it was stated before, the sampling frequency should be higher than twice the maximum frequency, or fs ≥ 2W , therefore: Signalling Rate ≥ 2N W

2.7

(2.29)

Quantization

The sampled version of the analog signal contains voltage levels which has to be scaled to predefined quantities in quantization phase. Each quantity is given a certain binary code in the encoding process. After those two processes finished, a binary sequence will represent the digital form of the analog input. This sampling, quantization and encoding are the main parts of any ADC. In the encoding process, a sequence of bits are assigned to different quantization values. Since there are a total of N = 2 n quantization levels, n bits are sufficient for the encoding process. In this way, we have n bits corresponding to each sample; since the sampling rate is fs samples/sec, we will have a bit rate of R = n × fs bits/sec.

2.7.1

Uniform Quantization

Basically, the quantizer will assign specific values for discrete samples of the analog signal. The quantized levels are discrete values step size  is the difference between each two consecutive values. There are many types of quantizer, while in this course only midriser type is covered. For midriser, when the quantizer output is rising will be zero when the input is zero, and it is up-rise from −/2 to /2 directly as shown in f igure 2.12,. this is why it is called midriser quantizer. It is important to understand that the x − axis of a continuous signal is the time domain t, and it will be replaced by nTs in the discrete domain after sampling. On the other hand, the ADC designer has to choose the number of bits v to represent the digital form of the signal. Let x(nTs ) is a continuous signal that its peak voltage ranges from −xmax to xmax . The quantizer will map those voltages to 0 q 0 levels on the vertical axis as shown in figure 2.12. Hence: Total amplitude range = xmax − (−xmax ) = 2xmax

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