By R. Kuc

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**Example text**

The following geometric relationships can be derived from Fig. 12) Arg[c] = arctan[bla]. 14) When both a and ro are real numbers, each element of the sequence is expressed in complex vector notation with magnitude e"" and phase ron. A special case of this sequence occurs when~ a = O. ----I:--~ Real FIGURE 2-6 The complex plane. -______ ~ Real F1GURE 2·7 Interpretation of the Euler identities in the complex plane. relationships shown in Fig. 7, which are known as the Euler identities. 17) The property of sinusoidal sequences, that all values can be represented with ro in the range [-Jr, Jr), is also true for complex exponentials.

T(O){h(lI)} II ... r. oI II + + •• • Ji'lGURE U The input sequence to a \inear system can be viewed as the superposition in time of a set of scaled unit-sample sequences. The output sequence is then the superposition of the scaled unit-sample responses. 3. veneer. Let y(n) =lx(n + 1) + lx(n) + lx(n -1) To find the unit-sample response {h(n, O}}, we set {x(n}} = {d(n}} and solve for the values of y(n) for each n: For n = -2, y( -2) = id( -1) + Id( -2) + ld( -3) = O. Forn S -2, y(n) = 0, since d(n) = 0 for n <0.

This result is also true for the sine sequence. COMPLEX EXPONENTIAL SEQUENCE. Before we consider the complex exponential sequence, we will establish the notation for complex numbers and the complex plane used throughout this book. Let c be a complex number. 10) where a is the real part, b is the imaginary part, and j = (_1)112. The representation of c on the complex plane is shown in Fig. 6. 11) lei is the magnitude and Arg[c] is the argument or phase, which is 24 INTRODUcnON TO DIGITAL SIGNAL PROCESSING {cos (0.