9Complex Signals – Quadrature Filters – Interpolators
Complex signals in the form of sets of complex numbers are currently used in digital signal analysis. Some examples of these sets are presented in the chapters on discrete Fourier transforms. In this chapter, analytic signals – a particular category of complex signal – will be studied. Such signals exhibit some interesting properties and occur primarily in modulation and multiplexing. The properties of the Fourier transforms of real causal sets will be examined first [1–3].
9.1 The Fourier Transform of a Real and Causal Set
Consider a set of elements x(n) whose Z-transform is written as:
The Fourier transform of this set is obtained by replacing Z with ej2πf in X(Z):
If the elements x(n) are real numbers, we obtain:
The values of X(f) at negative frequencies are complex conjugates of the values at positive frequencies. The supplementary condition of causality can be imposed on the set x(n) and the consequences for X(f) will now be examined.
The function X(f) can be separated into real and imaginary parts:
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