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L23 315 F11 EE 315 Fall 2011 SAMPLING THEOREM PROOF Introduction: The sampling theorem for sequences is valid for sequences that are ideally band limited to 1/(2n); the Fourier transform has a cycle that is identically zero for frequencies larger than 1/(2n). Such band-limited sequences can be recovered from equally spaced samples that are separated by the integer n. Thus the amount of storage required to store such a sequence can be reduced by a factor of n. The sampling theorem is represented in the block diagram below. The input is to an ideal low pass filter that eliminates any aliasing frequency components in the input signal. If the input sequence is ideally band limited, the output of this filter, is equal to the input sequence, . The second system in the cascade merely outputs every n th component of the filtered input sequence: The third system in the cascade, the up sampler, merely replaces the deleted components of the filtered input, sfin [∗] , with zeroes. This zero padded sequence is then input to the initial filter. The output of this cascade of four systems is equal to the filtered input, . Block Diagram: Analysis of the System: 1

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L23 315 F11EE 315Fall 2011

SAMPLING THEOREM PROOF

Introduction:The sampling theorem for sequences is valid for sequences that

are ideally band limited to 1/(2n); the Fourier transform has a cycle that is identically zero for frequencies larger than 1/(2n). Such band-limited sequences can be recovered from equally spaced samples that are separated by the integer n. Thus the amount of storage required to store such a sequence can be reduced by a factor of n.

The sampling theorem is represented in the block diagram below. The input is to an ideal low pass filter that eliminates any aliasing frequency components in the input signal. If the input

sequence is ideally band limited, the output of this filter, is equal

to the input sequence, . The second system in the cascade merely outputs every nth component of the filtered input sequence:

The third system in the cascade, the up sampler, merely replaces the deleted components of the filtered input, sfin [∗], with zeroes. This zero padded sequence is then input to the initial filter. The output of this cascade of four systems is equal to the filtered

input, .

Block Diagram:

Analysis of the System: This system is best understood in the frequency domain; and is

easiest to understand with visual plots of the Fourier transforms of the sequences involved.

The first Fourier transform is plotted below; since Fourier transforms are complex valued in general, a two dimensional “face” represents this three dimensional nature of the Fourier transform; the shading of the eyes at small frequency magnitudes represents that the Fourier transform is different at positive and negative frequencies. When the sequence is real, the Fourier transform at a negative

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frequency equals the conjugate of the corresponding positive frequency.

The input is drawn to represent the potential presence of frequency components at aliasing frequencies. For this visualization, the down sampling factor is two, n=2. In the cycle of the periodic Fourier transform, the aliasing components are at frequencies with magnitudes between ¼ and ½. For a general down sampling factor of n, the aliasing components are at frequency magnitudes greater than 1/(2n).

The Anti-Aliasing Filter:

Since the transfer function of the anti-aliasing filter is, in general, given as below: the transfer function of a cycle of the anti-aliasing filter is zero when the magnitude of the frequency is greater than1/(2n). The output of the initial anti-aliasing filter is ideally band limited to a centered band of width 1/n.

Since the cycle of the transfer function is identically zero at frequencies above 1/(2n), the cycle of the Fourier transform is also zero at frequencies with magnitudes above 1/(2n). The sequence at the output of the anti aliasing filter is ideally band limited to 1/(2n). The visualization of the output of the anti aliasing filter follows.

The unit pulse response of this ideal low pass filter can be easily evaluated with symbolic Matlab; such a Matlab script follows.

Symbolic Matlab Script:>> syms f k n real;l=int(exp(i*2*pi*f*k),f,-1/n/2,1/n/2) <l=sin((pi*k)/n)/(pi*k)> >> zero=simplify(l-(sinc(k/n))/n) <zero=0>

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The unit pulse response of this ideal low pass filter is

.

The Down Sampler:

The second system in the cascade of four systems is the down sampler where the sampling rate is reduced by a factor of n; the down sampler output only contains every nth sample of the input sequence, sfin[*]=fin[n*]. The relation between the input and output Fourier transforms of a down sampler was shown earlier to be:

The Fourier transform at the output of the down sampler, by a factor of n, is scaled in amplitude by (1/n), stretched in frequency by a factor of n, and is repeated periodically with period unity.

The visualization of the Fourier transform of the output of the down sampler has been scaled in amplitude by 1/n and stretched by a factor of n to fill the entire band of frequencies. The periodic repetition of the Fourier transform does nothing to the shape of the Fourier transform since it is periodic with unity period. The amplitude of the Fourier transform is merely multiplied by n; however, it had already been divided by n. The net effect on the amplitude of the Fourier transform is to leave the amplitude unchanged at A.

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The up sampler is defined in the frequency domain; the output Fourier transform is merely the input Fourier transform compressed by a factor

of n: S⃗FINS(⋅)=⃗SFIN(n⋅). The down sampler output Fourier transform is thus periodic with period ½; the figure that follows visualizes the down sampler output. The low frequency portion of this Fourier transform is identical to the output of the anti aliasing filter; if an anti aliasing filter is used to filter the down sampler output the filter output will be equal to the output of the anti aliasing filter.

In some sense, this anti aliasing filter serves to interpolate for the zero

A visualization of the sampling system output Fourier transform follows; It is identical to the output of the anti aliasing filter; when the

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system input is ideally band limited the input to the sampling system is then equal to the output of the sampling system.

The numeric Matlab script below illustrates the manner in which the final filter interpolates the up sampled signal. The unit pulse response of the final filter; the down sampling factor is n=3. The unit pulse

response of the filter is upr [∗]=sinc (∗¿ n); It is assumed there are only three non zero components in the up sampler:

sfins3 [∗]={ 1 ,∗¿03 ,∗¿32 ,∗¿6

0 , otherwise

Numeric Matlab Script:

>> n=3;sfins=[1 3 2];>> dt=.01;t=-10:dt:10;s1=sfins(1)*sinc(t./n);s2=sfins(2)*sinc((t-3)./n);

s3=sfins(3)*sinc((t-6)./n);>> plot(t,s1,':',t,s2,':',t,s3,':',t,s1+s2+s3),hold on,plot(3*(0:2),sfins,'*'),

xlabel('*,t'),ylabel('sfins(*),sfins(t)'),title(n+3)

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