- Discrete-Time Fourier Transform

Abstract

The discrete-time Fourier transform has essentially the same properties as the continuous-time Fourier transform, and these properties play parallel roles in continuous time and discrete time. As with the continuous-time Four ier transform, the discrete-time Fourier transform is a complex-valued function whether or not the sequence is real-valued. Furthermore, as we stressed in Lecture 10, the discrete-time Fourier transform is always a periodic function of fl. If x(n) is real, then the Fourier transform is corjugate symmetric, which implies that the real part and the magnitude are both even functions and the imaginary part and phase are both odd functions. Thus for real-valued signals the Fourier transform need only be specified for positive frequencies because of the conjugate symmetry. Whether or not a sequence is real, specification of the Fourier transform over a frequency range of 2 7r specifies it entirely. For a real-valued sequence, specification over the frequency range from, for example, 0 to a is sufficient because of conjugate symmetry. The time-shifting property together with the linearity property plays a key role in using the Fourier transform to determine the response of systems characterized by linear constant-coefficient difference equations. As with continuous time, the convolution property and the modulation property are of particular significance. As a consequence of the convolution property, which states that the Fourier transform of the convolution of two sequences is the product of their Fourier transforms, a linear, time-it variant system is represented in the frequency domain by its frequency response. This representation corresponds to the scale factors applied at each frequency to the Fourier transform of the input. Once again, the convolution property can be thought of as a direct consequence of the fact that the Fourier transform decomposes a signal into a linear combination of complex exponentials each of which is an eigenfunction of a linear, time-invariant system. The frequency response then corresponds to the eigenvalues. The concept of filtering for discrete-time signals is a direct consequence of the convolution property. The modulation property in discrete time is also very similar to that in continuous time, the principal analytical difference being that in discrete time the Fourier transform of a product of sequences is the periodic convolution