As a preview of things to come, note that one signal 4.15 is projected onto another signal using an inner product. The inner product computes the coefficient of projection4.16 of onto . If (a sampled, unit-amplitude, zero-phase, complex sinusoid), then the inner product computes the Discrete Fourier Transform (DFT), provided the frequencies are chosen to be . For the DFT, the inner product is specifically
Another case of importance 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
If, more generally, (a sampled complex sinusoid with exponential growth or decay), then the inner product becomes
Why have a transform 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 growing functions of time such as growing 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 a polynomial in . As such, it can be fully characterized (up to a constant scale factor) by its zeros in the plane. Similarly, the transform of an exponential can be characterized to within a scale factor by a single point in the plane (the point which generates the exponential); since the transform goes to infinity at that point, it is called a pole of the transform. More generally, the transform of any generalized complex sinusoid is simply a pole located at the point which generates the sinusoid. Poles and zeros are used extensively in the analysis of recursive digital 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. This topic will be taken up in detail in Book II .
In the continuous-time case, we have the Fourier transform which projects onto the continuous-time sinusoids defined by , and the appropriate inner product is
Finally, the Laplace transform is the continuous-time counterpart of the transform, and it projects signals onto exponentially growing or decaying complex sinusoids: