Discrete Time Fourier Transform and Its Extension to z ‐Transforms

Summary

This chapter extends the Fourier series representation of discrete time periodic functions to aperiodic functions. We investigate the power of the discrete time Fourier transform in manipulating the frequency content of discrete time signals. We study the basic properties of discrete time Fourier transforms, such as linearity, time shifting, time scaling, and difference properties. We observe many similarities between the continuous and discrete time Fourier transforms. As in the continuous time Fourier transforms, the energy of a discrete time signal is also preserved in the continuous frequency domain. We also study several duality properties between the time and frequency domain. We observe that difference equations become algebraic equations in the frequency domain. Thus, solving them in the frequency domain is rather easier than solving them in the time domain. We also show that there is a one‐to‐one correspondence between the representation of LTI systems by impulse response, frequency response, and difference equations.