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Investigation of different aspects of filtering in the frequency domain, particularly the use of discrete Fourier transforms.

Because we are interested in actual computations rather than analytic calculations, we must consider the detailsof the discrete Fourier transform. To compute the length- N DFT, we assume that the signal has a duration less than or equal to N . Because frequency responses have an explicit frequency-domain specification in terms of filter coefficients, we don't have a direct handle on whichsignal has a Fourier transform equaling a given frequency response. Finding this signal is quite easy. First of all, notethat the discrete-time Fourier transform of a unit sample equals one for all frequencies. Since the input and output oflinear, shift-invariant systems are related to each other by Y 2 f H 2 f X 2 f , a unit-sample input, which has X 2 f 1 , results in the output's Fourier transform equaling the system's transfer function .

This statement is a very important result. Derive it yourself.

The DTFT of the unit sample equals a constant (equaling 1). Thus, the Fourier transform of the output equals thetransfer function.

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In the time-domain, the output for a unit-sample input is known as the system's unit-sample response , and is denoted by h n . Combining the frequency-domain and time-domain interpretations of alinear, shift-invariant system's unit-sample response, we have that h n and the transfer function are Fourier transform pairs in terms of the discrete-time Fourier transform .

h n H 2 f

Returning to the issue of how to use the DFT to perform filtering, we can analytically specify the frequency response,and derive the corresponding length- N DFT by sampling the frequency response.

k k 0 N 1 H k H 2 k N
Computing the inverse DFT yields a length- N signal no matter what the actual duration of the unit-sample responsemight be . If the unit-sample response has a duration less than or equal to N (it's a FIR filter), computing the inverse DFT of the sampled frequencyresponse indeed yields the unit-sample response. If, however, the duration exceeds N , errors are encountered. The nature of these errors is easily explained byappealing to the Sampling Theorem. By sampling in the frequency domain, we have the potential for aliasing in the time domain(sampling in one domain, be it time or frequency, can result in aliasing in the other) unless we sample fast enough. Here, theduration of the unit-sample response determines the minimal sampling rate that prevents aliasing. For FIR systems —they by definition have finite-duration unit sample responses — the number of required DFT samples equals theunit-sample response's duration: N q .

Derive the minimal DFT length for a length- q unit-sample response using the Sampling Theorem. Because sampling in thefrequency domain causes repetitions of the unit-sample response in the time domain, sketch the time-domain resultfor various choices of the DFT length N .

In sampling a discrete-time signal's Fourier transform L times equally over 0 2 to form the DFT, the corresponding signal equals the periodic repetition of the original signal.

S k i s n i L
To avoid aliasing (in the time domain), the transform length must equal or exceed the signal's duration.

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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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