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 [13].

9.1 The Fourier Transform of a Real and Causal Set

Consider a set of elements x(n) whose Z-transform is written as:

upper X left-parenthesis upper Z right-parenthesis equals sigma-summation Underscript n equals minus infinity Overscript infinity Endscripts x left-parenthesis n right-parenthesis upper Z Superscript negative n

The Fourier transform of this set is obtained by replacing Z with ej2πf in X(Z):

upper X left-parenthesis f right-parenthesis equals sigma-summation Underscript n equals minus infinity Overscript infinity Endscripts x left-parenthesis n right-parenthesis e Superscript minus normal j Baseline 2 Baseline Superscript pi italic n f

If the elements x(n) are real numbers, we obtain:

(9.1)upper X left-parenthesis negative f right-parenthesis equals ModifyingAbove upper X left-parenthesis f right-parenthesis With bar

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:

(9.2)upper X left-parenthesis f right-parenthesis equals upper X Subscript upper R Baseline left-parenthesis f right-parenthesis plus j normal upper X Subscript upper I Baseline left-parenthesis f right-parenthesis

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