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*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Why (Generalized) Complex Sinusoids are Important

As a preview of things to come, note that one signal

^{5.10}isprojectedonto another signal using aninner product. The inner product computes thecoefficient of projection^{5.11}of onto . If (a sampled, unit-amplitude, zero-phase, complex sinusoid), then the inner product computes theDiscrete Fourier Transform(DFT), provided the frequencies are chosen to be . For the DFT, the inner product is specifically

Another commonly used case is the

Discrete Time Fourier Transform(DTFT) which is like the DFT, except that the transform accepts an infinite number of samples instead of only . In this case, frequency is continuous, and

The DTFT is what you get in the limit as the number of samples in the DFT approaches infinity. The lower limit of summation remains zero because we are assuming all signals are zero for negative time. This means we are working withunilateralFourier transforms. There are also correspondingbilateraltransforms for which the lower summation limit is .If, more generally, (a sampled complex sinusoid withexponential growth or decay), then the inner product becomes

and this is the definition of thetransform. It is a generalization of the DTFT: The DTFT equals the transform evaluated on theunit circlein the plane. In principle, the transform can also be recovered from the DTFT by means of “analytic continuation” from the unit circle to the entire plane (subject to mathematical disclaimers which are unnecessary in practical applications since they are always finite).Why have a tranform when it seems to contain no more information than the DTFT? It is useful to generalize from the unit circle (where the DFT and DTFT live) to the entire complex plane (the transform’s domain) for a number of reasons. First, it allows transformation of

growingfunctions of time such as unstable exponentials; the only limitation on growth is that it cannot be faster than exponential. Secondly, the transform has a deeper algebraic structure over the complex plane as a whole than it does only over the unit circle. For example, the transform of any finite signal is simply apolynomialin . As such, it can be fully characterized (up to a constant scale factor) by itszerosin the plane. Similarly, the transform of anexponentialcan be characterized by a single point of the transform (the point whichgeneratesthe exponential); since the transform goes to infinity at that point, it is called apoleof the transform. More generally, the transform of anygeneralized complex sinusoidis simply apolelocated at the point which generates the sinusoid. Poles and zeros are used extensively in the analysis ofrecursivedigital filters. On the most general level, every finite-order, linear, time-invariant, discrete-time system is fully specified (up to a scale factor) by its poles and zeros in the plane.In the

continuous-timecase, we have theFourier transformwhich projects onto the continuous-time sinusoids defined by , and the appropriate inner product is

Finally, the

Laplace transformis the continuous-time counterpart of the transform, and it projects signals onto exponentially growing or decaying complex sinusoids:

The Fourier transform equals the Laplace transform evaluated along the “ axis” in the plane, i.e., along the line , for which . Also, the Laplace transform is obtainable from the Fourier transform via analytic continuation. The usefulness of the Laplace transform relative to the Fourier transform is exactly analogous to that of the transform outlined above.