ABSTRACT
In Chapter 2 we observed that the DTFT https://www.w3.org/1998/Math/MathML" display="inline"> X ( e j ω ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003337164/b3f22258-c274-4568-80b9-bec8a50b02ca/content/eq312.tif"/> of a sequence https://www.w3.org/1998/Math/MathML" display="inline"> x ( n ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003337164/b3f22258-c274-4568-80b9-bec8a50b02ca/content/eq313.tif"/> is continuous with respect to frequency. The continuous spectrum is made up of an infinite number of samples that requires infinite memory to store and infinite time to process. It is not feasible to process such a signal in a DSP processor. The solution to this problem is to process a finite number of samples taken from https://www.w3.org/1998/Math/MathML" display="inline"> X ( e j ω ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003337164/b3f22258-c274-4568-80b9-bec8a50b02ca/content/eq314.tif"/> . The number of samples taken is normally made equal to the number of samples of the sequence https://www.w3.org/1998/Math/MathML" display="inline"> x ( n ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003337164/b3f22258-c274-4568-80b9-bec8a50b02ca/content/eq315.tif"/> . Such samples form what is referred to as the Discrete Fourier Transform (DFT) which is defined only for finite length sequences as follows:
