By R.D. Richtmyer

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7 CT exponential function A CT exponential function, with complex frequency s = σ + jω0 , is represented by x(t) = est = e(σ +jω0 )t = eσ t (cos ω0 t + j sin ω0 t). 38) The CT exponential function is, therefore, a complex-valued function with the following real and imaginary components: Re{est } = eσ t cos ω0 t; real component Im{est } = eσ t sin ω0 t. imaginary component Depending upon the presence or absence of the real and imaginary components, there are two special cases of the complex exponential function.

D) Time-shifted version x [−15 − 3k] of (c). −6 −4 −2 0 2 4 6 8 10 k 12 −6 −4 −2 0 2 4 6 8 10 k 12 (d) Solution Express x[−15 – 3k] = x[−3(k + 5)] and follow steps (i)–(iii) as outlined below. (i) Compress x[k] by a factor of 3 to obtain x[3k]. The resulting waveform is shown in Fig. 31(b). (ii) Time-reverse x[3k] to obtain x[−3k]. The waveform for x[−3k] is shown in Fig. 31(c). (iii) Shift x[−3k] towards the left-hand side by five time units to obtain x[−3(k + 5)] = x[−15 − 3k]. The waveform for x[−15 – 3k] is plotted in Fig.

P1: RPU/XXX P2: RPU/XXX CUUK852-Mandal & Asif QC: RPU/XXX May 25, 2007 T1: RPU 18:7 32 Part I Introduction to signals and systems 6 4 6 4 2 0 2 0 −2 −4 −6 −30 −2 −4 −6 −30 −20 −10 0 10 k 30 20 (a) −20 −10 0 10 20 k 30 (b) Fig. 17. 05k)u[k]. (a) Real component; (b) imaginary component. 10 CT unit impulse function The unit impulse function δ(t), also known as the Dirac delta function† or simply the delta function, is defined in terms of two properties as follows: (1) amplitude δ(t) = 0, ∞ t = 0; δ(t)dt = 1.