14.1 The Sampling Theorem
The result of applying the methods described previously results in data which are a far cry from the quantity to be measured, a continuous function of time referred to as the signal. The data generally departs from this ideal in two respects. Firstly the value of the signal at a particular time is represented by a binary coded integer, which is not continuous by its very nature. This limitation is variously described as a quantizing error or by the concept of resolution. In essence, the data is uncertain by the equivalent of 1/2 in its integer representation. Thus if digitizing a sample leads to the integer 500, the precision of the measurement is 0.1%.
Secondly, measurements of the signal are only made at a discrete set of times, which are usually spaced at equal intervals, Ts. Thus the continuous function v(t) is relaced by an array of the form
where tn = (n-1)Ts, if one insists in defining the first sample to be at t = 0. In the above v0 is the ADC quantizing voltage. Provided the integer representing the sample is sufficiently large, the quantizing error is not the limiting factor so the digitizing of the data will be overlooked in the following considerations.
The dimension of the array is limited by the size of the memory allocated for storage of the samples. The sampling period is generally limited by the time required for the analogue-to-digital conversion to which must be added the time for storage in memory. Thus there are intervals of time between consecutive samples for which data does not exist, and this limitation deserves some consideration.
The discrete-time signal which results from the sampling procedure arises from the periodic enabling of a linear gate by a logic signal which is a repetitive series of narrow time intervals for which it is at logic one and the gate is open. If the gate signal is denoted by g(t), the linear gate output may be written in terms of the continuous input signal as
The gating function may be idealized as being at logical 1 for a negligible duration, ie as consisting of a series of equally spaced, infinitely narrow spikes, giving the gating function
This gives
Of course the sampled data only exists over a finite range of n. The infinite extension is justified if the signal vanishes or is insignificant in the regions not included in the sampling.
Since the gating function is periodic it can be represented by a Fourier series of the form
where ws=2p/Ts is the angular sampling frequency. Thus the Fourier transform of the gating function is given by
The representation of the discrete time function in the frequency domain follows from the convolution theorem. This states, it will be recalled, that convolution and multiplication are conjugate operations. Thus if V(w) and Vd(w) are the Fourier transforms of the continuous and discrete time signals respectively
Thus the transform of the discrete time signal consists of an infinite set of replicates of the transform of the continuous time signal. The replicates are related by translations through an integral number of sampling frequencies. An important case is that of a band-limited signal. In this case there exists a maximum frequency beyond which the transform vanishes.
Such a situation is illustrated in
the accompanying figure. The shaded areas represent the regions of frequency space occupied by the transform of
the discrete time signal, and for each replicate extend an amount wm around
a central frequency which is an integral number of sampling frequencies. Only the two neighbouring replicate regions
beginning at zero of the infinite set are shown. Provided ws > 2wm, the occupied regions do not overlap. If the discrete signal is then filtered
by the uniform low pass filter shown the transform of the continuous time signal will be recovered. Hence in this
situation it is possible to completely reconstruct the continuous time signal from the discrete time signal. The
only remaining limitation with respect to a sampled signal is the quantizing error discussed above. The reconstruction
in the time domain corresponds to convolution with the transform of H(w), a function
of the general form sinc(x) = sinx/x. This amounts to a special interpolation procedure which yields the value
of the signal at any point. This result is referred to as the (Nyquist) sampling theorem, and the minimum sampling
frequency required is called the Nyquist limit.
Sampling at a frequency below the Nyquist limit results in overlapping of the replicate transforms and the sample is said to be undersampled. The contributions from the neighbouring transform corresponding to frequency w in the continuous transform now appear as contributions at ws- w, a phenomenon known as aliasing.
14.2 Repetitive Signals
The concepts used for sampling can also be applied to the situation of signals consisting of pulses repeated at regular intervals. If the pulse shape is p(t) then the signal can be written
where g is the gate function of Eqn(14.3) with the sampling interval Ts replaced by the repetition interval Tr. This follows from the fact that convolution of a function with d(t-t) shifts the function by t. The convolution theorem then states that the frequency representation of the signal may be calculated as the product of the transform of the pulse with the transform of the gate function. The result of this operation yields
Since the right most term vanishes except for the infinite set of discrete frequencies w = nwr Eqn(14.9) can be written as
Now suppose the signal were processed by a low pass filter with H(w)=0 for w>=wr. The filtered transform Yf would consist only of the n=0 term,
The filtered signal in the time domain would become
Note that this signal is in fact a constant, ie independent of time, described of course as DC. This result is also reflected in the nature of the transform, Eqn(14.11) which is a line spectrum at zero frequency. Finally, one has
so that the DC level is
an entirely reasonable result. There are of course many applications of these results. In AC to DC conversion the rectification process can be looked upon as creating a repetitive series of pulses corresponding approximately to one half of a sinusoid. In practice the low pass filter remains finite for the remaining terms in Eqn(14.10), so that the filtered signal does have time varying components at the harmonics of amplitude
which contribute ripple to the output.
The result in Eqn(14.14) provides a means of frequency to voltage conversion. If a standardized logic pulse is generated from a periodic input signal level crossing then the constant pulse area can effectively be calibrated to unity in volt-sec. The DC level resulting from low pass filtering would equal f=1/Tr volts. Such a system could also be considered as an analogue ratemeter, since f corresponds to the pulse rate. It is a remarkable fact which requires much more detailed consideration, that this result also applies to pulses which are randomly spaced. In this case there is no unique repetition time. Such signals arise from convolution with a function of the form Sd(t-tn) where the tn are statistically distributed. Such signals are said to arise from random point processes, such as occur in the detection of radiation events. The process is described by an event rate, corresponding to the inverse of the average interval between events. The event rate plays the role of frequency, so the filtered output is directly proportional to the event rate. Of course the output pulses from a radiation detector are themselves statistical with regard to amplitude and shape, and again a trigger circuit must be used to produce a logic pulse.