Theory:

 

Contents:

 

♦ Nyquist sampling theorem

♦ Aliasing

♦ Sampling of non-bandlimited signal (anti-aliasing filter)

♦ Signal reconstruction

♦ Zero-order hold

♦ First-order hold

♦ The discrete Fourier transform

♦ Picket-fence effect

♦ Zero padding

♦ Properties of discrete Fourier transform

 

 

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 Sampling:

 

A continuous time signal can be processed by processing its samples through a discrete time system. For reconstructing the continuous time signal from its discrete time samples without any error, the signal should be sampled at a sufficient rate that is determined by the sampling theorem.

 

 Nyquist Sampling Theorem:

 

If a signal is band limited and its samples are taken at sufficient rate then those samples uniquely specify the signal and the signal can be reconstructed from those samples. The condition in which this is possible is known as Nyquist sampling theorem and is derived below.

A real signal whose spectrum is bandlimited to D Hz  [X(f) = 0 for | f |>D] can be reconstructed from its samples taken uniformly at a rate f_s > 2D samples/sec. We can say the minimum sampling frequency is f_s=2D Hz.

Consider a signal x(t) as shown in Fig.1 and its spectrum is shown in Fig.2 which is noted to be bandlimited to D=1 Hz . Sampling x(t) at a rate of f_s Hz, (means f_s samples/sec) can be mathematically represented as multiplying x(t) by an impulse train consisting of unit impulses repeating periodically every T seconds, where T= 1/ f_s. The resulting sampled signal x'(t) is shown in Fig. 3. The spectrum of the sampled signal is shown in Fig. 4.

 

Fig.1  Signal x(t)		Fig.2  Spectrum of signal x(t)

                                                                                      

Fig.3 Sampled signal		Fig.4 Spectrum of sampled signal

 

The sampled signal consists of impulses spaced every T seconds. The n^th impulse, located at t = nT, has a strength x(nT), the value of x(t) at t = nT,

                                                                     ..............(1)

Since the impulse train is a periodic signal of period T, it can be expressed as a trigonometric Fourier series

                                         ................(2)  

 where                              

Therefore       .........(3)

To find X^'(f), the Fourier transform of x'(t), we take the Fourier transform of the right hand side of the above equation, term by term. The transform of the first term in the bracket of the above equation (3) is X(f). Similarly the transform of the second termis. This represents spectrum X(f) shifted toandand so on to infinity. This result means that the spectrum X^'(f) consists of X(f) repeating periodically with periodrad/s, or f_s =1/T Hz, as shown in Fig.4. Therefore

                                                                          

If we want to reconstruct x(t) from x'(t), we should be able to recover X(f)  from X^'(f). This recovery is possible if there is no overlap between successive cycles of X^'(f). From Fig.4 we noted that for no overlap between successive cycle of X^'(f) we should have

                                                              or 

Thus as long as the sampling frequency f_s is greater than twice the signal bandwidth D (in Hz), X^'(f) will consist of nonoverlapping repetitions of X(f). In such a case it indicates that x(t) can be recovered from its samples x'(t) by passing the sampled signal x'(t) through an ideal low pass filter of bandwidth D Hz. If a signal is sampled with a sampling frequency f_s which is below the highest frequency present in the signal then the signal is called undersampled signal and if the signal is sampled with a sampling frequency f_s which is slightly higher than twice the bandwidth of the signal or highest frequency present in the signal then the signal is called oversampled signal.

 

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Aliasing:

 

In reconstructing a signal from its samples, there is another practical difficulty. The sampling theorem was proved on the assumption that the signal x(t) is bandlimited. All practical signals are time limited, i.e., they are of finite duration. As a signal cannot be timelimited and bandlimited simultaneously. Thus, if a signal is timelimited, it cannot be bandlimited and vice versa (but it can be simultaneously non timelimited and non bandlimited). Clearly it can be said that all practical signals which are necessaily timelimited, are non bandlimited, they have infinite bandwidth and the spectrum X^'(f) consists of overlapping cycles of X(f) repeating every f_s Hz (sampling frequency). Because of infinite bandwidth, the spectral overlap will always be present regardless of what ever may be the sampling rate chosen. Because of the overlapping tails, X^'(f) has not complete information about X(f) and it is not possible, even theoretically to recover x(t) from the sampled signal x'(t). If the sampled signal is passed through an ideal lowpass filter, the output is not X(f) but a version of X(f) distorted as a result of some causes:

 

• The loss of the tail of X(f) beyond |f | > f_s/2  Hz.

 

• The reappearance of this tail inverted or folded onto the spectrum. The spectra cross at frequency f_s/2 = 1/2T  Hz. This frequency is called the folding frequency. The spectrum folds onto itself at the folding frequency. For instance, a component of frequency (f_s/2)+ f_x shows up as or act like a component of lower frequency (f_s/2)- f_x in the reconstructed signal. Thus the components of frequencies above f_s/2 reappear as components of frequencies below f_s/2. This tail inversion is known as spectral folding or aliasing which is shown in Fig. 5. In this process of aliasing not only we are losing all the components of frequencies above f_s/2 Hz, but these very components reappear as lower frequency components. This reappearance destroys the integrity of the lower frequency components.

 

Fig.5  Aliased signal

 

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Sampling of Non-bandlimited Signal: Anti-aliasing Filter

 

Anti aliasing filter is a filter which is used before a signal sampler, to restrict the bandwidth of a signal to approximately satisfy the sampling theorem. The potential defectors are all the frequency components beyond f_s/2 Hz. We should have to eliminate these components from x(t) before sampling x(t). As a result of this we lose only the components beyond the folding frequency f_s/2 Hz. These frequency components cannot reappear to corrupt the components with frequencies below the folding frequency. This suppression of higher frequencies can be accomplished by an ideal filter of bandwidth f_s/2 Hz. This filter is called the anti-aliasing filter. The anti aliasing operation must be performed before the signal is sampled. The anti aliasing filter, being an ideal filter is unrealizable. In practice, we use a steep cutoff filter, which leaves a sharply attenuated residual spectrum beyond the folding frequency f_s/2.

 

 Signal Reconstruction:

 

The process of reconstructing a continuous time signal x(t) from its samples is known as interpolation. In the sampling theorem we saw that a signal x(t) band limited to D Hz can be reconstructed from its samples. This reconstruction is accomplished by passing the sampled signal through an ideal low pass filter of bandwidth D Hz. As seen in the above equation(3), the sampled signal contains a component and to recover x(t) or X(f), the sampled signal must be passed through an ideal lowpass filter having bandwidth D Hz and gain T.

The description of the process of reconstruction in the frequency domain is to find the DTFT of the discrete-time signal, change the variable F→f / f_s , multiply by T and find the inverse CTFT. We find the DTFT from the samples using

                                                                          

Here we are using the notations X(f) for CTFT and X(F) for DTFT.

 

Fig. 6 DTFT of sinc function

 

For sinc function, the DTFT is illustrated in Fig.6. To isolate the function indexed by k = 0, we can multiply the DTFT by a rectangle function that is wide enough to include the k = 0 alias but not wide enough to include any other aliases. So the corner of the rectangle must be at a value of F which is greater than F_m= f_m/ f_s, where f_m is the highest frequency present in the signal x(t) but less than 1 - F_m or more compactly, F_m< F_c<1 - F_m. The isolated k = 0 replica is then

                                                                

Next we make the change of variable F→ f / f_s and multiply it by T 

                                                   

Now we have to find out the inverse CTFT, which is

                                                               

 Let F_c = f_c/f_s , then 

                                                              

By exchanging the order of integration and summation we can recognize the integrand as a time shifted inverse CTFT of a frequency domain rectangle function.

                                                             

Then x[n] = x(nT),                         

The reconstruction process consists of replacing each sample by a sinc function, centered at the time of the sample and scaled by the sample value x(nT) times 2f_c/ f_s and adding all the functions so created. Suppose the signal is sampled at exactly Nyquist rate f_s= 2f_m, Then f_m= f_s/2 = f_s- f_m and F_m= 1/2 = 1- F_m.

The requirement  F_m< F_c< 1- F_m cannot be met, in this case we must allow F_c= F_m, which means that f_c= f_m= f_s/2 This will work till the signals spectrum does not have an impulse at  f_m. (If there is an impulse at f_m , it will be aliased in the sampling process). In this limiting case, the interppolation is described by the simpler expression

                                                               

Interpolation consists of simply of multiplying each sinc function by its corresponding sample value and then adding all the scaled and shifted sinc functions as shown in Fig.7 given below.

Fig.7 Weighted sinc function.

 

The interpolated value at any point is the sum of contributions from infinitely many weighted sinc functions.

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 Zero-order hold:

 

There are many techniques that can be used to reconstruct a signal and the selection of the technique to be used is depends on what accuracy of reconstruction is required and how oversampled the signal is. Probably the simplest approximate reconstruction idea is to simply let the reconstruction always be the value of the most recent sample as shown in Figure 8.

                               

                    Fig.8 Zero order hold signal reconstruction.             Fig.9 Impulse response of a Zero order hold.

 

It is a simple technique because the samples in the form of numerical codes, can be the input signal to a Digital to Analog converter, which is clocked to produce a new output signal with every clock pulse. This technique produces a signal which has a stair step shape that follows the original signal. This type of signal reconstruction can be modeled (except for quantization effects) by passing the impulse sampled signal through a system called a zero order hold whose impulse response is

                                      

which is shown in Figure 9.

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First-order hold:

 

One popular way of further reducing the effects of the aliases is to follow the zero order hold with a practical lowpass filter that smooths out the steps caused by the zero order hold. The zero order hold causes a delay to the original signal because it is causal.

Another natural reconstruction idea is to interpolate between samples with straight lines as shown in Fig.10 below. This is obviously a better approximation of the original signal but it is a little harder to implement.

 

                                 

 Fig.10 Signal reconstruction by straight-line interpolation.   Fig.11 Impulse response of causal first-order hold.

 

 

As shown in above figure the value of the interpolated signal at any time depends on the value of the previous sample and the value of the next sample and this is not possible in real time because the value of the next sample is not known in real time. But if we delay the reconstructed signal by one sample time T, we can make the reconstruction process causal and the reconstructed signal would appear as like the Figure 12.

 

                                                            

                                       Fig.12 Straight line signal reconstruction delayed by one sample time

 

This interpolation can be accomplished by following the zero order hold by an identical zero order hold. This means that the impulse response of such a signal reconstruction filter would be the convolution of the zero order hold impulse response with itself

                                              

    

This type of filter is called first order hold. Its transfer function is. This transfer function is similar to that of a zero order hold but attenuates aliases more because its magnitude diminishes faster with increasing frequency.

 

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The discrete Fourier transform (DFT):

If x(kT) and X(rf₀) are the k^th and r^th samples of x(t) and X(f) respectively, then we can define new variables x_k and X_r as

                                                                       

 and  X_r = X(rf₀)   where                                 

We will now see that x_k and X_r are related by the following equation

  DFT,                                                                             

 and  IDFT                                                                  

Here X_r is the discrete Fourier transform of 'x_k' and 'x_k' is the IDFT of X_r . The notation  x_k ⇔ X_r  is also used to indicate that x_k and X_r are a DFT pair. Remember that x_k is T₀/ N₀ times the k^th sample of x(t) and X_r is the r^th sample of X(f).

 

 Picket fence effect:

 

The numerical computation method yields only the uniform sample values of X(f) or x(t). Using this method is like viewing a signal and its spectrum through a "picket fence". The major peaks of X(f) or x(t) could lie between two samples and may remain hidden, there is a situation giving a false picture of the reality. Such misleading results can be avoided by using a sufficiently large 'N₀' the no. of samples, which increases the resolution.

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 Zero padding:

 

Observing X_r is like observing the spectrum X(f) through a "picket-fence". If the frequency sampling interval f₀ is not sufficiently small, we could miss out some significant details and obtain a misleading picture. To obtain a higher number of samples, we need to reduce f₀. Because f₀ = 1/T₀, a higher number of samples requires us to increase the value of 'T₀' the period of repetition for x(t). This option increases 'N₀' the number of samples of x(t), by adding samples with a value of 0. This addition of dummy samples is known as zero padding. zero padding increases the number of samples and it may help in getting a better idea of the spectrum X(f) from its samples X_r.

 

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 Properties of discrete Fourier transform:

 

The DFT is basically the Fourier transform of the sampled signal which is repeated periodically. So the properties of the Fourier transform can also applied for DFT.

 1. Linearity property:          If     and   then

 2. Time shifting (Circular shifting):                 

 3. Frequency shifting:                                         

 4. Circular convolution:                                     and   

 For two 'N₀' periodic sequences x_k and 'g_k' the circular convolution is defined as

                                                                      

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