Filtering in the Frequency Domain 2.3 2000/08/10 2004/08/09 13:35:25.843 GMT-5 Don Johnson dhj@rice.edu Melissa Selik mselik@alumni.rice.edu Don Johnson dhj@rice.edu DFT difference equation filtering Describes the use of the discrete Fourier transform for filtering signals in the frequency domain. 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 response might 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 frequency response indeed yields the unit-sample response. If, however, the duration exceeds N , errors are encountered. The nature of these errors is easily explained by appealing 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, the duration 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 the unit-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 the frequency domain causes repetitions of the unit-sample response in the time domain, sketch the time-domain result for 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. Express the unit-sample response of a FIR filter in terms of difference equation coefficients. Note that the corresponding question for IIR filters is far more difficult to answer: Consider this example. The difference equation for an FIR filter has the form y n m 0 q b m x n m The unit-sample response equals h n m 0 q b m δ n m which corresponds to the representation described in a problem of a length q 1 For IIR systems, we cannot use the DFT to find the system's unit-sample response: aliasing of the unit-sample response will always occur. Consequently, we can only implement an IIR filter accurately in the time domain with the system's difference equation. Frequency-domain implementations are restricted to FIR filters. Another issue arises in frequency-domain filtering that is related to time-domain aliasing, this time when we consider the output. Assume we have an input signal having duration N x that we pass through a FIR filter having a length- q 1 unit-sample response. What is the duration of the output signal? The difference equation for this filter is y n b 0 x n … b q x n q This equation says that the output depends on current and past input values, with the input value q samples previous defining the extent of the filter's memoryof past input values. For example, the output at index N x depends on x N x (which equals zero), x N x 1 , through x N x q . Thus, the output returns to zero only after the last input value passes through the filter's memory. As the input signal's last value occurs at index N x 1 , the last nonzero output value occurs when n q N x 1 or n q N x 1 . Thus, the output signal's duration equals q N x . In words, we express this result as "The output's duration equals the input's duration plus the filter's duration minus one." Demonstrate the accuracy of this statement. The unit-sample response's duration is q 1 and the signal's N x . Thus the statement is correct. The main theme of this result is that a filter's output extends longer than either its input or its unit-sample response. Thus, to avoid aliasing when we use DFTs, the dominant factor is not the duration of input or of the unit-sample response, but of the output. Thus, the number of values at which we must evaluate the frequency response's DFT must be at least q N x and we must compute the same length DFT of the input. To accommodate a shorter signal than DFT length, we simply zero-pad the input: Ensure that for indices extending beyond the signal's duration that the signal is zero. Frequency-domain filtering, diagrammed in , is accomplished by storing the filter's frequency response as the DFT H k , computing the input's DFT X k , multiplying them to create the output's DFT Y k H k X k , and computing the inverse DFT of the result to yield y n .

To filter a signal in the frequency domain, first compute the DFT of the input, multiply the result by the sampled frequency response, and finally compute the inverse DFT of the product. The DFT's length must be at least the sum of the input's and unit-sample response's duration minus one. We calculate these discrete Fourier transforms using the fast Fourier transform algorithm, of course.