Tuesday, 5 June 2018

SAMPLING THEOREM


EDITOR: B. SOMANATHAN NAIR


1. INTRODUCTION
In analog-to-digital (A/D) conversion, analog signals are first sampled, then held, and finally converted into digital signals. Rate of sampling is computed based on the Nyquist sampling theorem, which states that:
 An analog signal can be recovered from its samples completely if and only if sampling is done under the condition that sampling frequency used is equal to, or greater than, twice the maximum frequency of that signal.
In mathematical form, the sampling theorem may be stated as

                                                                   fs  ³  2fm   (1)                                                              

where fs is the sampling frequency and fm is the maximum frequency of input analog signal.
            To illustrate the sampling theorem, consider the sampling of analog audio signals in the range of 100 Hz to 4 kHz. In this case, the maximum audio frequency fm = 4 kHz. Therefore the sampling frequency required is

                                 fs   ³  8 kHz                                                       

2. PROOF OF SAMPLING THEOREM
We find that sampling is a convolution operation. This operation may be expressed mathematically as

                                                             y(t) = x(t)p(t)   (2)                                                 

where x(t) is the input analog signal, p(t) is the sampling pulses, and y(t) is the  sampled output. The star symbol represents the convolution operation. Convolution in time domain can be seen to be equal to multiplication in frequency domain. In the Fourier-transform domain, we express the convolution related to sampling as

                                 Y(w) = X(w)P(w)   (3)                                   

where Y(w) is the Fourier-transform y(t), X(w) is the Fourier-transform of x(t), and  P(w) is the Fourier-transform of p(t). Figure 1 shows an example of a time-domain input signal x(t), and  Fig. 2 shows the frequency components X(w) in that signal, obtained by Fourier transformation of x(t). It may be noted here that x(t) is the waveform that we observe on an oscilloscope and X(w) is the pattern that we observe on a spectrum analyzer. From Fig. 2, we observe that the frequency components are discrete signals with each signal component having finite amplitude. We have drawn a blue-colored dotted line as the envelope covering the samples in the figure.
            As stated above, for sampling, we use an impulse- (delta-) function train p(t), defined as

                                                                p(t)  = d(t-nT)   (4)                                           

where n = 0, 1, 2, … , and T is any unit time period.


           



 In the case of sampling, we take Ts as the sampling interval, which is equal to 1/fs, where fs is the sampling frequency. The Fourier transform of the delta-function (impulse) train p(t) is given by

                                   
                  
where 2π/Ts = ws is the amplitude of the sampled pulses. From Eq. (5), we find that Fourier transform of a time-domain impulse train is a frequency-domain impulse train. Substituting Eq. (5) into Eq. (3) yields

                                                 
                        

Equation (6) may be expanded as

Y(w) = ws [X(w)d (w) + X(w)d(w -ws) +  X(w)d(w -2ws)  X(w)d(w -nws) +
  X(w)d(w +ws) + X(w)d( w +2ws)X(w)d(w +nws)]  (7)  

From Eq. (7), we find that sampling operation has produced a sample of order zero and an infinite number of side-band samples of orders 1, 2, 3, …, n. We also notice that the zeroth-order sample is located at w = 0, the first-order samples at w = ± ws, the second-order samples at w = ± 2ws, and so on. The amplitudes of all these samples are equal to ws [X(w)d(w)]. Figures 3, 4, and 5 give the pictorial representations of the sampling operation, where we have used a compressed version of the envelope (with a maximum frequency of wm) of X(w) for convenience.
In Fig. 3, as stated above, we find that the product function ws X(w) is located at w = ws, ± 2ws, ± 3ws, and so on. The shapes of envelope, shown in Fig. 3, as stated above, represent the product term wsX(w), and are arbitrarily chosen here [the actual shape of these is dependent on the actual shape of the term X(w)]. We also notice that the bandwidth of each of the samples is equal to 2wm, where wm is the maximum frequency of the sampled signal.
From Fig. 3, we find that the samples remain just touching each other, if ws = 2wm. In Fig. 4, we notice that the samples are isolated from each other and are not touching each other. This occurs for ws > 2wm. under these two conditions, the original signal may be recovered from the samples by using a low-pass filter with cut-off frequency ws = 2wm. it may be noticed that forws > 2wm, there exists a guard band between each of the samples; this helps in designing the low-pass filter because cut-off frequency is not sharp. In the figures, the sample corresponding to the original signal is shown in red lines. It is this sample that we have to recover. The rest of the samples to be neglected are shown in blue lines.   
Figure 5 shows the condition when ws < 2wm. Under this condition, the samples overlap each other and separating them becomes difficult. This situation is known as aliasing.
From the explanations given above, we conclude that original signal can be recovered from its samples if we sample the signal at the rate of ws ≥ 2wm. But this is the statement of the sampling theorem.














 














 













No comments:

Post a Comment

DISCRETE SIGNAL OPERATIONS

EDITOR: B. SOMANATHAN NAIR 1. INTRODUCTION In the previous two blogs, we had discussed operations of scaling and shifting on conti...