Fourier Series Representation of Discrete Time Periodic Signals

Summary

This chapter introduces the Fourier series representation of discrete time periodic functions. We represent discrete time periodic functions in a Hilbert space spanned by the orthogonal harmonically related complex exponential functions. The discrete nature of the signal enables us to replace the integral operation with the summation operation to compute the spectral coefficients of the signal, which are periodic and discrete time. They correspond to the coordinates of the discrete time periodic functions in Hilbert space, spanned by a finite number of harmonically related and orthogonal discrete time complex exponentials. Thus, discrete time functions can be represented in finite‐dimensional Hilbert spaces. In linear time‐invariant systems, the spectral coefficients of the output are just the scaled version of the spectral coefficients of the input, where the scale factor is the eigenvalue of the system, corresponding to the complex exponential eigenfunctions.