Analogue Pulse Modulation Schemes

Pulse Modulation Schemes

In analogue pulse modulation, the carrier is a a periodic pulse train and we may continuously vary the pulse amplitude, pulse width, pulse frequency or the position of each pulse in its time slot in proportion to the sampled messages of the message waveform. In each of these cases the informationn is carried in analogue form although the transmission takes place at discrete intervals of time. Examples of these modulation formats are illustrated in figure fig-pm-schemes.

Pulse Amplitude Modulation (PAM):
	This may be considered to be the pulse carrier equivalent of DSB-SC amplitude modulation although it is not exactly equivalent as may be seent when we look at the difference between Natural Sampling and Pulse Amplitude Modulation sampling.
Pulse Width Modulation (PWM):
	The width (duration) of each carrier pulse is varied according to the amplitude of the message signal. Signal recovery is easily acheived by low pass filtering.
Pulse Position Modulation (PPM):
	The position of each carrier pulse within its time slot of width $T$ is varied. Signal recovery is acheived by measurement of the time interval between this pulse and a regular clock pulse train operating at rate $1/T$ .
Pulse Frequency Modulation (PFM):
	This is directly equivalent to FM. The pulse repetition frequency is varried in porportion to the message signal samples; signal recovery is as for FM. PFM is used in low-cost, high performance analogue communications using fibre optics where nonlinearities in the transfer characteristics of optical sources impair the performance of PAM.

Nyquist Sampling Theorem

From what we have learn't in the tutorial on modulation basics we know that if we sample a signal at regular instants in time we are going to produce a periodic spectrum. Nyquists theorem is concerned with knowing how often we sample the signal to that we can reproduce it witout distortion. If we have a signal strictly bandlimited to bandwidth $W$ and at regular intervals seperated by $T$ seconds (rate $1/T$ ) then the resulting spectrum is going to be copys of the spectrum seperate by $1/T$ Hz as shown in figure fig-nyquist.

Note that because we are modulating the original signal we see its whole double sided spectrum of width $2W$ copied at frequency intervals of $1/T$ . Therefore we see that if $T \le 1/(2W)$ these multiple copies of the signal spectrum are completed seperated, and we could filter out one to regenerate the original signal without distortion. However if we sample at too low a rate, $T > 1/(2W)$ then we see that there is overlap between the copies in the sampled spectrum and there would be distortion in the recovered signal. Therefore we can state the requirements for sampling, the Nyquist Sampling Theorem as follows:

A band-limited signal of finite energy, which has no frequency components higher than $W$ Hz is completely described by sampling values of the signal at instants of time separated by $1/2W$ seconds.

If we do not satisfy this criterion we cannot recover the original signal. In particular we may see frequency components which do not exist in the original signal. This is called aliasing. For example if we have a 4 kHz input signal and sample at a rate of only 5kHz we will see frequency components at (5+4)=9kHz which is removed by a low pass filter and at (5-4)=1kHz which isn't i.e. we will see a 1kHz signal after low pass filtering a 4kHz signal sampled at only 5kHz.

An example, which you may see in the laboratory, is that when using a digital sampling oscilloscope which has too low a sample rate (i.e. with a low time base set) and we look at high frequencies. We may actually see low frequency aliased signals which are not present at the input to the oscilloscope. Care must therefore be taken when using such instruments - in particuar if the frequency of the signal you are measuring changes as you change the timebase (sample rate) then it is probably oscilloscope aliasing that you are seeing.

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Natural Sampling

In natural sampling a bandlimited signal $f(t)$ is multiplied by a rectangular pulse train $p(t)$ (the periodic gate function) with period $T$ (the sampling period) to give the sampled signal $f_s(t)=f(t) p(t)$ .

Information Signal for Natural Sampling

We start with the information signal which is smooth in time and has a limited bandwidth.

Gate Function for Natural Sampling

Our gate function is a rectangular pulse train which has a spectrum which is the sinc function evaluated at multiples of the repetition frequency.

Natural Sampling of a waveform in the time and frequency domains

Now we multiply these together (in the time domain). The resultant is the convolution of the two spectra in the frequency domain, which is particularly easy to evaluate due to discrete nature of the sampling spectra. It is the signal spectrum, repeted at multiples of the sampling frequency and multiplied by the amplitude of the sampling spectrum at those points. Natural sampling is characterised by sampling pulse tops which precisely follow the variations in $f(t)$ .

The original signal can be recovered simply by applying a low pass filter to select only the baseband component of the spectrum which is the same as the signal spectrum. It is clear that we do not need to use rectangular pulses at all as selection of the sampling pulse shape serves only to specify the shape of the envelope of $F_s(\omega)$ which won't affect the result after filtering.

Pulse Amplitude Modulation

In pulse amplitude modulation the sample pulses are flat topped, and so, in contrast with natural sampling, do not follow the variations of the signal being sampled. The digital circuit used to achieve PAM is referred to as sample and hold and it comprises two operations: the instantaneous sampling of the message signal $f(t)$ followed by lengthening the duration of each sample to a constant value $\tau$ . The value of $\tau$ is chosen to reduce the bandwidth requirements: if it is too short the transmission would require excessive bandwidth. It is important to note the distinction between this process and natural sampling.

The Discretely Sampled information signal

In the case of flat-topped sampling our input signal is the sampled at discrete points in time so its spectrum is the baseband spectrum repeated at multiples of the sampling frequency as shown in figure fig-flat-samplinga.

Rectangular Pulse for Flat topped Sampling

The next step in the process is the hold operation which amounts to passing these samples through a filter to acieive the required rectangular pulse shape - a filter with a rectangular impulse response $q(t)$ . This hold function is illustrated in figure fig-flat-samplingb. Its spectrum is $Q(\omega)$

Flat Topped Sampling of a waveform in the time and frequency domains

The output of this filter is the convolution of the samples with the rectangular hold function which corresponds to the multiplication of the two spectra as shown in figure fig-flat-samplingc.

This spectrum differs from the naturally sampled one, the PAM output spectrum is obtained by multiplying together two frequency functions and the original form of $F(\omega)$ is not maintained - the replicas are not true replicas of the signal spectrum.

If we were to pass this signal through a low pass filter as we did for the natural sampling case we would obtain the spectral output which is $F(\omega)$ . The way around this is to pass the recovered signal through another filter, with a transfer function $1/Q(\omega)$ . This is referred to as equalisation, and the additional filter is an equaliser. The importance of the hold time $\tau$ should now be clear. If we use a shorter hold time $\tau$ then its spectrum will broaden and become flatter and the distortion produced on the received spectrum would be reduced. It is found that if the ratio of sample time to sample period $\tau/T \leq 1$ , the maximum difference between the equaliser transfer function $1/Q(\omega)$ and the ideal low pass filter is <1% in which case an additional equalisation filter is usually considered unnecessary.

Why are flat topped pulse so important when a naturally sampled signal is recoverable without distortion by low pass filtering. The reason is that the need to preserve the sampled pulse shapes in transmission conflicts with the basic advantages of digital communications. In transmission over distances which require mid-path amplification, the effects of additive noise would make the natural sampling format no better than analogue. When the pulse shape is not important, as with flat-toped sampling used in PAM, repeaters may be use to regenerate rather than amplify the signal giving giving more robustness in the presence of noise.

Sample and Hold

If the sampling pulse is very much smaller than the sampling interval then the signal power at the output low pass filter (LPF) at the receiver may be very small, requiring large amounts of amplification. An efficient way around this is to use a sample-and-hold circuit at the receiver, as illustrated in figure fig-sample-and-hold. In this circuit the switched is closed for the duration of the sample pulse. If the source impedance $R_s$ is small the capacitor charges to the input voltage level within the sample time. Between samples the switch is opened. If the load resistance R is large the capacitor will retain its voltage between samples until the switch is closed again.

The output of the sample and hold circuit is therefore as shown: smoothing is obtained by passing the signal through a LPF. The sample and hold is a reliable and efficient PAM demodualtor with good noise immunity and removes the need for large amounts of equalisation. Note however that equlisation is required since the output of the sample and hold circuit is effectively long flat-topped sample pulses.

Time Division Multiplexing

Time Division Multiplexing Operation

With relatively short sampling pulses PAM signals occupy only a small fraction of the sampling interval, and room is left between pulses into which samples from other signals can be insereted. This technique of combining pulses from $N$ independant channels in a definite time sequence is referred to as time division multiplexing (TDM) . The principle can be applied to most pulse modulation formats. It is illustrated in figure fig-tdm-concept for PAM signals.

A Time Division Multiplexed Signal

Each of the $N$ message signals is band limited by passing it through a low-pass filter (LPF). The filtered signals are passed to a commutator, which, in practice, constructed from high speed digital switches but can bethought of as a rotating switch which sequentially switched to the output of each LPF. The commutator takes a narrow sample of each of the $N$ signals at the sampling rate $f=1/T$ at least equal to twice the LPF bandwidth as required by the sampling theorem. It also interleaves the $N$ samples within the sampling interval $T$ . In figure fig-tdm-signal the result of this process is shown for 2 signals. In the PAM case the TDM output pulse train so obtained is transmitted; for other pulse modulation formats a further pulse modulator may be used. At the receiving end of the channel the TDM signal passes through another commutator which sequentially distributes the pulses to each of $N$ outputs where each message is recovered by low pass filtering. The second commutator must operate in complete synchronism with the first.

Note that the channel bandwidth $B_c$ required to pass the TDM signal must also satisfy the Nyquist condition. Thus the $N$ interleaved pulse trains constitute a single pulse train with separation $T_s=T/N$ . The sampling theorem requires $B_c \geq 1/2T_s$ in order to prevent information loss. The TDM signal can, in effect, be filtered to $B_c$ (low pass filtered) yet still permit separation of the constituent messages by resampling at the receiver using, for example, the sample and hold circuit (See fig-sample-and-hold).

PPM and PWM Generation

Both pulse position modulation (PPM) and pulse width modulation (PWM) signals can be generated using the scheme shown here.

Pulse Position and Pulse Width Modulator

Pule Position and Pulse Width Modulation Generation

The input signal is fed through a sample and hold circuit. Each of the samples is added to a sawtooth waveform generated synchronously to produce the sum waveform shown in figure fig-ppm. This waveform is sent to a comparator with a threshold set, as indicated by the blue line on the sum waveform. The incoming amplitude , when added to the sawtooth, will cause a change in the time where the sum waveform crosses the threshold. Therefore the the amount of time spend above threshold will depend on the signal amplitude, and we have pulse width modulation. With PWM, the information is actually contained in the relative positions of the pulse edges. Consequently, the longer pulses expend significant amounts of power which is not carrying information. The PPM signal is produced by triggering a pulse generator on the falling edge of each PWM pulse to generate a pulse of constant width. Note that although PPM is generally an efficient analogue pulse modulation scheme it does require a local generation of the clock timing since, in contrast to PAM and PWM which carry clearly recognisable clock timing, the timing is lost.

Since the PPM system is band limited, it must have a finite rise time and this must place uncertainty in the determination of the input signal which must be proportional to the rise time of the system, inversely proportional to its bandwidth. The uncertainty in the system is referred to as the resolution. Note, the resolution only has a meaning in a system subject to noise and distortion since otherwise a given point on the rise time characteristic will always correspond to the same relative timing instant. Generally the accepted criterion for specifying the resolution of a PPM system equates to the rise time. It follows that any two pulses in a PPM train must always be separated by at least the width of the system impulse response.