1. Module 3
Pulse Modulation

2. Contents:- Sampling theorem , Re-construction theorem & Aliasing
Natural & Flat top Sampling
Pulse Modulation Schemes –
PAM
PWM
PPM
Multiplexing Techniques
FDM
TDM

3. Sampling…
In brief its - a small part or quantity of something (sample), intended to show what the whole thing is like.
The sampling process should not yield any loss of the information.

4. Sampling The sampler takes a snapshot of the x(t) for every Ts
Analog
signal
Discrete-time
sequence

5. Sampling: Time Domain
Many signals originate as continuous-time signals, e.g. conventional music or voice
By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbers
n  {…, -2, -1, 0, 1, 2,…}
Ts is the sampling period. Ts t Ts
s(t)
Sampled analog waveform
impulse train

6. Sampling of a sinusoid

7. Sampling: Frequency Domain
Sampling replicates spectrum of continuous-time signal at integer multiples of sampling frequency
Fourier series of impulse train where ws = 2 p fs
Modulation by cos(s t)
Modulation by cos(2 s t) w
G(w) ws
2ws
-2ws
-ws
F(w)
2pfmax
-2pfmax

11. Aliasing
If sampling theorem is not fulfilled (it is not fs > 2fmax ), neighboring replicas of the original spectrum overlap and the overlapped parts of the spectrum differ from the corresponding parts of the original spectrum. This phenomenon is called aliasing.
If aliasing occurs, the original spectrum cannot be found and the original analog signal cannot therefore be ideally reconstructed.
To combat aliasing we need to have –
Prior to sampling a low pass anti-aliasing filter to remove unwanted HF components of the signal.
Signal is sampled at a rate slightly higher than Nyquist rate.

12. Sampling Theorem
A continuous-time signal x(t) with frequencies no higher than fmax can be reconstructed exactly from its samples x[n] = x(n Ts) if the samples are taken at a rate fs = 1/Ts which is greater than 2 fmax. Where, fmax refers to the maximum frequency component in the signal that has significant energy.
Consequence of violating sampling theorem is corruption of the signal in digital form.
So, we have -
Nyquist rate = 2 fmax
Nyquist interval = 1/fs seconds.
What happens if fs = 2fmax?

13. Reconstruction theorem
Consider a continuous-time signal xc(t). This signal is sampled with sampling interval T to form the discrete-time signal x[n] = xc(nT). The reconstruction theorem states that,
as long as xc(t)was appropriately sampled (faster than the Nyquist rate), xc(t)can be exactly reconstructed from the samples x[n].

14. Method of Reconstruction
Reconstruction is a two-step processes. The first step is to form a continuous-time representation of the sampled signal x[n]. We convert the sequence x[n] to a continuous- time impulse train xs(t).
Sampling a continuous signal creates, in the frequency domain, periodic repetitions of the frequency response of the original signal. The periodic repetitions occur at multiples of the sampling frequency where fs is the sampling frequency and fm is the bandwidth of xm(t).
In the second step of reconstruction, we apply a low-pass filter hr(t) to remove the unwanted frequencies created by the sampling process. According to the reconstruction theorem, if hr(t) is designed appropriately, the output of this filter is exactly equal to xm(t).

15. Reconstruction …

16. Filtering…

17. Reconstruction filter response

18. Ideal Signal Reconstruction

19. Sinc Function
It could be considered as an impulse response of an ideal low pass filter with pass band magnitude response 1/2 fm and bandwidth fm. The function has its peak value at the origin and goes through 0 at integer multiples of bit duration T.

20. Practical Signal Reconstruction

21. Sampling Altogether…

22. Why 44.1 kHz for Audio CDs? Sound is audible in 20 Hz to 20 kHz range:
fmax = 20 kHz and the Nyquist rate 2 fmax = 40 kHz
What is the extra 10% of the bandwidth used?
Rolloff from passband to stopband in the magnitude response of the anti-aliasing filter
Okay, 44 kHz makes sense. Why 44.1 kHz?
At the time the choice was made, only recorders capable of storing such high rates were VCRs.
NTSC: 490 lines/frame, 3 samples/line, 30 frames/s = samples/s
PAL: 588 lines/frame, 3 samples/line, 25 frames/s = samples/s

23. Sampling
As sampling rate increases, sampled waveform looks more and more like the original
Many applications (e.g. communication systems) care more about frequency content in the waveform and not its shape.

24. Natural Sampling

25. Flat top Sampling

26. Pulse Modulation…
Pulse modulation involves communication using a train of recurring pulses.
Two types of pulse modulation –
Analog : A periodic pulse train is used as carrier and some characteristics of each pulse (amplitude, position or duration) is changed in a continuous manner in accordance with the sample value of the message signal. So information is transmitted in analog form but in discrete times.
Digital: Message signal is represented in a form with is discrete both in amplitude and time, thereby allowing the transmission of it in digital form as a sequence of coded pulses.
There are several pulse modulation techniques
Pulse Amplitude Modulation
Pulse Width Modulation
Pulse Position Modulation

27. Pulse Amplitude Modulation (PAM)
In PAM, the amplitudes of regularly spaced pulses are varied in proportion to the corresponding sampled 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 rectangular pulse varies with the message signal, whereas in PAM it is maintained at.

28. Generation of PAM
There are two operations involved in the generation of PAM signal:
Instantaneous sampling of the message signal m(t) every Ts seconds, where the sampling rate fs = 1/Ts is chosen in accordance with the sampling theorem.
Lengthening the duration of each sample so obtained to some constant value T. By lengthening each pulse duration to some time T we can avoid using excessive bandwidth as its inversely proportional to the pulse duration.

29. PAM
PAM is rather stringent in its system requirement, such as short duration of pulse.
Also, the noise performance of PAM may not be sufficient for long distance transmission.
Accordingly, PAM is often used as a mean of message processing for time-division multiplexing, from which conversion to some other form of pulse modulation is subsequently made.
Distortion caused by PAM in transmitting flat top pulses for an analog message signal is called aperture effect.

30. Other forms of Pulse Modulation
Pulse-duration modulation(PDM), also referred to as Pulse-width modulation (PWM), where samples of the message signal are used to vary the duration of the individual pulses in the carrier.
The maximum analog signal amplitude produces the widest pulse, and the minimum analog signal amplitude produces the narrowest pulse. Note, however, that all pulses have the same amplitude.

31. PPM
Pulse-position modulation(PPM), where the position of a pulse relative to its un modulated time of occurrence is varied in accordance with the message signal.
The higher the amplitude of the sample, the farther to the right the pulse is positioned within the prescribed time slot. The highest amplitude sample produces a pulse to the far right, and the lowest amplitude sample produces a pulse to the far left.

32. Figures …

34. Comparisons between PDM and PPM
PPM is more power efficient because excessive pulse duration consumes considerable power.
Final note
It is expected that PPM is immune to additive noise, since additive noise only perturbs the amplitude of the pulses rather than the positions.
However, since the pulse cannot be made perfectly rectangular in practice (namely, there exists a non-zero transition time in pulse edge), the detection of pulse positions is somehow still affected by additive noise.