1.1

EXPERIMENT 1

SAMPLING AND PULSE MODULATION

1.0 OBJECTIVES

Description in time and frequency domain of:

• delta function pulse train

• rectangular pulse train

• bandlimited pulse train

• pulse amplitude modulation (PAM)

- instantaneous (delta function) sampling

- natural sampling

- flat-top sampling

• pulse-width modulation (PWM)

• pulse-position modulation (PPM)

• demodulation of PAM, PPM, PWM

2.0 THEORY

2.1 Pulse Trains

In this section we describe three kinds of pulse trains in both the time and frequency

domains:

• delta function pulses

• rectangular pulses

• bancllimited rectangular pulses

1.2

A train of delta function pulses (Fig. 1a) may be written in the time domain as

00

(1.1 )n=-oo

and in the frequency domain (Fig. 1b) as

6TsU) =; f 6(1 - nIs)s n=-oo

(1.2)

where Is = l/Ts.

A rectangular pulse of finite length T and finite amplitude A may be written as

{A O~t~T

r(t) 0 otherwise (1.3)

A train c( t) of rectangular pulses spaced at intervals Ts (Fig. 2a) with T < Ts may be

written in the time domain as

00

c(t) = L r(t - nTs) (1.4 )n=-oo

Expanding as a Fourier series, we may write

(X) TA 00

c(t) = L Cn exp(j27rnIst) = T L sinc(nIsT) exp(j7rnIs(t - T))n=-(X) s n=-(X)

(1.5 )

where sinc(x) = sin(7rx)7rX

Thus in the frequency domain (Fig. 2b)

TA 00

CU) = T L sinc (nfsT) exp( -j7rnfsT)6U - nfs)5 n=-oo

(1.6)

1.3

where fsT = TITs is the pulse duty cycle. The signal power IC(fW is plotted in Fig. 2b.

The exponential delay term exp( - j7rnfsT) in (1.5) and (1.6) may be eliminated if the

pulse T(t) is defined for It I <A/2 instead of as in (1.3).

A real pulse train which can be generated by practical equipment will be bandlimited,

i.e. passed through a lowpass filter H(f) with cutoff frequency B. A train of bandlimited

rectangular pulses may be written in the frequency domain as (Fig. 3)

S(f) = C(f)H(f) (1.7)

The pulse rise time tr may be defined as the time for the pulse to rise from 10% to 90%

of its value:

tr = K I B, (1.8)

where K depends upon the characteristics of the filter H(f) (number of poles, filter type).

2.2 PAM Signals

A PAM signal s(t) is the product of a message signal g(t) and a pulse train c(t). (Thus

for PAM the carrier c( t) is a pulse train, whereas for DSBSC the carrier is a sine wave). In

PAM the amplitude of the pulses is varied in step with the message signal. We distinguish

three types of sampling: instantaneous, natural, and fiat top, each with its own type of pulse

train c(t)

2.2.1 Instantaneous Sampling

If the pulse train c(t) = 6Ts(t) is a series of delta functions, then we have ideal or

instantaneous sampling and we may write the sampled signal s(t) in the time domain

00

s(t) ~ g6(t) = g(t)6Ts(t) = L g(nTs)6(t - nTs) (1.9)n=-oo

1.4

and in the frequency domain

1 00

G6(f) = T L G(f - nIs)S n=-oo

(1.10)

where Is = l/Ts is the sampling frequency.

Fig. 4 illustrates a message signal g(t) (speech, music) of bandwidth W.

Fig. 5 illustrates the corresponding PAM signal g6(t), G8(f) with a sampling rate Is =

l/Ts greater than the Nyquist rate 21;\1.

Note that the spectrum G8(f) contains replicas of G(f) centred at frequencies nIs.

Fig. 6 illustrates what happens when Is < 2W and the replicas overlap. Low frequency

components of the first replica are mixed in with high frequency components of the baseband

message spectrum. This effect is called aliasing. The original message signal g( t) may be

recovered using an ideal reconstruction filter H (f) of bandwidth W only if there is no aliasing.

2.2.2 Natural Sampling

A rectangular pulse train may be written in the time domain as Eqn. (1.4) and in the

frequency domain as Eqn. (1.6).

A practical PAM signal s(t) using a rectangular pulse train and natural sampling may

be written in the time domain as

s(t) = g(t)c(t)

l.e.

00

(1.11)n=-oo

1.5

and in the frequency domain

TA 00

S(f) = T L G(f - nfs)sinc(nfsT) exp( -j7rnfsT)s n=-oo

(1.12)

The delay term exp(j7rnfsT) arises because the center of the pulse r(t) is defined at

t = T/2 (Eqn. 1.3). This term would disappear if r(t) were defined over It I < T/2.

Fig. 7 illustrates a "natural sampling" PAM signal s(t), S(f) for a message signal of

bandwidth W.

Note that each replica G(f-n/Ts) is uniformly attenuated by a factor K(n) = sinc (nfsT)

depending only on n.

The PAM signal is demodulated to recover the message signal g(t) by passing s(t) through

an ideal low-pass filter of bandwidth B with W < B < (l/Ts) - W.

2.2.3 Flat-top Sampling

A PAM signal using flat-top sampling (sample-and-hold) is illustrated in Fig. 8 and may

be written in the time domain as

(X)

s(t) = L g(nTs)T(t - nTs) (1.13)n=-(X)

and in the frequency domain as

TA 00

S(f) = T L G(f - nfs)sinc (fT) exp( -j7r fT)s n=-(X)

(1.14)

Compare (1.10) with (1.12) and (1.12) with (1.14).

Fig. 8 illustrates a "flat-top sampling" PAM signal s(t), S(f) for a message signal of

uandwidth W.

1.6

Note that the spectrum of the message signal and all of the replicas are attenuated by the

factor sinc(fT) which depends on f. Thus the spectrum of the message signal is altered or

distorted by the sinc function. This "aperture distortion" or "sin x/x distortion" is caused

by the finite length of the sample and hold pulses, and becomes more severe if the pulse duty

cycle is increased.

The PAM signal may be demodulated to recover the message signal get) by passing set)

through an ideal low-pass filter of bandwidth B such that 1;\1< B < (l/Ts) - W, followed

by an x / sin x equalizer.

2.3 Pulse Time Modulation (PWM and PPM) Signals

In PAM, PWM and PPM the samples of the message signal are used to vary the ampli-

tude, width and position of the pulses respectively, as illustrated in Fig. 9.

PWM and PPM may be generated as illustrated in Fig. 10. The sum of a message signal

and a ramp signal are applied to a comparator or slicer. The slicer output is a PWM signal.

Note that with this method the pulse width depends on the message signal sampled at the

time of occurrence of the trailing edge. Thus the sampling interval Ts is not constant.

An alternative method to obtain PWM is to sample the message signal at constant

intervals Ts (just like PAM) and then digitally generate a pulse of width corresponding to

the sample value.

PPM may be obtained by generating a pulse on the trailing or falling edge of a PWM

signal. The width of this pulse is constant and arbitrary, but is normally chosen to be very

short.

The spectrum of PWM and PPM is complicated [3]. PWM and PPM may be demodu-

lated to recover the message signal using a low pass reconstruction filter (Fig. 11a), but the

message will be distorted by crossmodulation terms in the message bandwidth. A hard lim-

iter or slicer may be used prior to the low pass filter in order to reduce the noise. A PPM

signal may be demodulated using a properly synchronized clock signal and a flip-flop (Fig.

11b).

3.0 PREPARATION

3.1 Verify, (1.10), (1.12), (1.14).

1.7

3.2 Show that if AT = 1 and T -+ 0, then (1.14)-+ (1.10)

3.3 Perform the spectrum calculations in procedures 5.1.1 and 5.2.1

3.4 Write an expression for a PPM and PWM signal in the time domain, using unit stepand/ or rectangular functions.

4.0 EQUIPMENT

Spectrum AnalyzerOscilloscopeLab- Volt Power SupplyDigital MultimeterLab- Volt Dual Func. Gen.Lab-Volt RF /Noise Gen.Lab- Volt Low Pass Audio FilterLab-Volt PAM/ASK Gen.Lab- Volt PAM/ASK ReceiverLab- Volt PWM/PPM Gen.Lab- Volt PWM/PPM ReceiverLab-Volt FM/PM Receiver

5.0 PROCEDURE

#9401

#9402#9406#9426#9440#9441#9442#9443#9415

In section 5.1 we look at the spectrum of a pulse train and investigate the effect of the

duty cycle on the spectrum. Section 5.2 looks at the spectrum of a PAM signal, the effect

of the duty cycle and the differences between natural and flat-top sampling. In section 5.3

we see what happens if a signal is sampled at and below the Nyquist rate. Section 5.4 shows

the use of PAM on a real radio signal, and section 5.5 shows the time waveforms and spectra

of PWM and PPM signals.

5.1 Pulse Trains

5.1.1 Set the oscilloscope for 1:1 probes. Do not connect scope and meter to the circuit at

the same time. Calculate and measure the spectrum in the range 0 to 400 kHz for a 10

1.8

KHz pulse train with duty cycle of 10 and 50%. Pulse trains of varying duty cycle can

be generated with the Lab-Volt dual function generator. Set the spectrum analyzer to

[FREQ] [LINEAR]. Record the waveforms in both the time and frequency domains.

Compare the calculated and measured values of the spectral peaks in each case. The

linear display is easier to interpret.

5.1.2 Describe the qualitative change in the spectrum as the duty cycle is varied from 10 to

90%.

5.2 PAM Signals

5.2.1 Calculate and measure the spectrum to 300 KHz for the following PAM signals. Use

the test setup in Fig. 12.

message signal: 8 KHz sine wave (10Vp-p)

sampling frequency: 64 KHz (A=5V)

duty cycle: 25% (optional also 50%)

sampling type: natural, flat-top

Carefully adjust the message signal frequency and/or the sampling frequency until the

sampled signal is "frozen" on the oscilloscope display.

Record the waveforms in both time and frequency domain. Compare the calculated

and measured spectrum.

5.2.2 How does the duty cycle affect the spectrum?

5.2.3 How does fiat-top versus natural sampling affect the spectrum?

5.3 Aliasing

1.9

5.3.1 Record the spectrum to 200 KHz for the following PAM signals. Use the test setup in

Fig. 13.

message signal: 1 KHz pulse10% duty cyclefiltered by 4th order filter with an 8 KHz cut-off1.0 Vpk at input of low-pass filter

sampling frequency: 64 KHz, 10 Vp-p

type: natural sampling

5.3.2 From the observed spectrum predict the lowest sampling frequency which can be used

without aliasing.

5.3.3 Reduce the sampling frequency about 25% below this frequency and record the spec-

trum. Explain your observation.

5.4 PAM Demodulation

5.4.1 Use the test set up in Fig. 14.

Tune the FM receiver to a local station. Use the 300 ohm antenna.

Use the audio output of the FM receiver as the message signal for PAM.

Specify the lowest sampling frequency that may be used without aliasing. Choose duty

cycle of 50%. Choose the type of sampling such that there is no sin x/x distortion. 0b-

serve, sketch and comment on the resulting PAM waveform and spectrum. (Optional:

try 10% duty cycle also).

5.4.2 Demodulate the PAM waveform. Listen to the recovered message signal and comment

on the quality.

Reduce the sampling rate until distortion is apparent. Record the sampling frequency.

Describe the distortion.

1. 10

Introduce sin x/x distortion. Comment on the change in received message quality.

5.5 PWM and PPM

Do either 5.5.1 or 5.5.2.

5.5.1 FWM

Use the test setup in Fig. 15.

Generate a pulse width modulation signal using:

- Sampling Signal - Ramp wave, fs=30 KHz, lOVp-p

- Message Signal - Sine wave fo=3KHz, 2Vp-p

- Again, carefully adjust the frequency until the sampled signal is "frozen" on the oscil-

loscope display.

Sketch and explain the PWM time waveform and spectrum to 100 KHz.

Demodulate the PWM signal and record the recovered time waveform.

5.5.2 PPM

Repeat 5.51 for a PPM waveform using the same sampling and message signals. Adjust

the PPM receiver for a stable display.

6.0 REVIEW QUESTIONS

6.1 How does the envelope of the spectrum of a pulse signal give an indication of the pulse

duty cycle?

6.2 If speech is to be sampled at 8 KHz for digital transmission, what must the cutoff

frequency of the lowpass filter be to prevent aliasing?

6.3 Why can PWM and PPM signals be demodulated directly by low-pass filtering?

6.4 Explain how PPM can be used to transmit a DC voltage level in an ac coupled channel.

1.11

7.0 REFERENCES

[1.J S. Haykin, "Communication Systems", Wiley, 1983, Chapter 7.

[2.J P.Z. Peebles, "Digital Communication System", Prentice-Hall, 1987, Chapter 2.

[3.J M. Schwartz, W.R. Bennett, S. Stein, "Communication Systems and Techniques",

McGraw-Hill, 1966, pp. 248-255.

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1.15

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1.18

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1.19

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1.20

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