The Sampling Theorem and signal reconstruction

Any real signal will be transmitted along some form of channel which will have a finite bandwidth. As a result the received signal's spectrum cannot contain any frequencies above some maximum value,

. However, the spectrum obtained using the Fourier method described in the previous section will be characteristic of a signal which repeats after the interval, T. This means it can be described by a spectrum which only contain the frequencies, 0 (d.c.),

, where N is the largest integer which satisfies the inequality

. As a consequence we can specify everything we know about the signal spectrum in terms of a d.c. level plus the amplitudes and phases of just N frequencies — i.e. all the information we have about the spectrum can be specified by just 2N +1 numbers. Given that no information was lost when we calculated the spectrum it immediately follows that everything we know about the shape of the time domain signal pattern could also be specified by just 2N +1 values.

For a signal whose duration is T this means that we can represent all of the signal information by measuring the signal level at 2N +1 points equally spaced along the signal waveform. If we put the first point at the start of the message and the final one at its end this means that each sampled point will be at a distance

from its neighbours. This result is generally expressed in terms of the Sampling Theorem which can be stated as: ‘If a continuous function contains no frequencies higher than Hz it is completely determined by its value at a series of points less than apart.’

Consider a signal,

, which is observed over the time interval,

, and which we know cannot contain any frequencies above

. We can sample this signal to obtain a series of values, x, which represent the signal level at the instants,

, where i is an integer in the range 0 to K . (This means there are

samples.) Provided that

, where N is defined as above, we have satisfied the requirements of the Sampling Theorem. The samples will then contain all of the information present in the original signal and make up what is called a Complete Record of the original.

In fact, the above statement is a fairly ‘weak’ form of the sampling theorem. We can go on to a stricter form:

‘If a continuous function only contains frequencies within a bandwidth, B Hertz, it is completely determined by its value at a series of points spaced less than

seconds apart.’

This form of the sampling theorem can be seen to be true by considering a signal which doesn't contain any frequencies below some lower cut-off value,

. This means the values of

for low n (i.e. low values of

) will all be zero. This limits the number of spectral components present in the signal just as the upper limit,

, means that there are no components above

. This situation is illustrated in figure 7.2.

From the above argument a signal of finite length, T, can be described by a spectrum which only contains frequencies,

. If the signal is restricted to a given bandwidth,

, only those components inside the band have non-zero values. Hence we only need to specify the

values for those components to completely define the signal. The minimum required sampling rate therefore depends upon the bandwidth, not the maximum frequency. (Although in cases where the signal has components down to d.c. the two are essentially the same.)

The sampling theorem is of vital importance when processing information as it means that we can take a series of samples of a continuously varying signal and use those values to represent the entire signal without any loss of the available information. These samples can later be used to reconstruct all of the details of the original signal — even recovering details of the actual signal pattern ‘in between’ the sampled moments. To demonstrate this we can show how the original waveform can be ‘reconstructed’ from a complete set of samples.

The approach used in the previous section to calculate a signal's spectrum depends upon being able to integrate a continuous analytical function. Now, however, we need to deal with a set of sampled values instead of a continuous function. The integrals must be replaced by equivalent summations. These expressions allow us to calculate a frequency spectrum (i.e. the appropriate set of

values) from the samples which contain all of the signal information. The most obvious technique is to proceed in two steps. Firstly, to take the sample values,

, and calculate the signal's spectrum. Given a series of samples we must use the series expressions

to calculate the relevant spectrum values. These are essentially the equivalent of the integrals, 7.10 and 7.11, which we would use to compute the spectrum of a continuous function. The second step of this approach is to use the resulting

and

values in the expression

to compute the signal level at any time, t, during the observed period. In effect, this second step is simply a restatement of the result shown in expression 7.9. Although this method works, it is computationally intensive and indirect. This is because it requires us to perform a whole series of numerical summations to determine the spectrum, followed by another summation for each

we wish to determine. A more straightforward method can be employed, based upon combining these operations. Expressions 7.12 and 7.13 can be combined to produce

which, by a fairly involved process of algebraic manipulation, may be simplified into the form

where the Sinc function can be defined as

and

is the time interval between successive samples.

Given a set of samples,

, taken at the instants,

, we can now use expression 7.15 to calculate what the signal level would have been at any time, t, during the sampled signal interval.

Clearly, by using this approach we can calculate the signal value at any instant by performing a single summation over the sampled values. This method is therefore rather easier (and less prone to computational errors!) than the obvious technique. Figure 7.2 was produced by a BBC Basic program to demonstrate how easily this method can be used.

Although the explanation given here for the derivation of expression 7.15 is based upon the use of a Fourier technique, the result is a completely general one. Expression 7.15 can be used to ‘interpolate' any given set of sampled values. The only requirement is that the samples have been obtained in accordance with the Sampling Theorem and that they do, indeed, form a complete record. It is important to realise that, under these circumstances, the recovered waveform is not a ‘guess' but a reliable reconstruction of what we would have observed if the original signal had been measured at these other moments.

Summary

You should now be aware that the information carried by a signal can be defined either in terms of its Time Domain pattern or its Frequency Domain spectrum. You should also know that the amount of information in a continuous analog signal can be specified by a finite number of values. This result is summarised by the Sampling Theorem which states that we can collect all the information in a signal by sampling at a rate

, where B is the signal bandwidth. Given this information we can, therefore, reconstruct the actual shape of the original continuous signal at any instant ‘in between’ the sampled instants. It should also be clear that this reconstruction is not a guess but a true reconstruction.

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