Introduction to signals - Continuous and Discrete Time Signals and Systems - page 31

6 6

4 4

2 2

0 0

−2 −2

−4 −4

k k

−6 −6

−30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30

(a) (b)

Fig. 1.16. DT complex
exponential function x [ k ] =
exp( j0.2π k - 0.05 k ). (a) Real
component; (b) imaginary
component.

is obtained by expanding

k = γ k ,

e ( σ + j 0 )

x [ k ] =

(1.40)

where γ = ( σ + j Ω 0 ) is a complex number. Equation (1.40) is more compact
than Eq. (1.39).

1.2.9 Causal exponential function

In practical signal processing applications, input signals start at time t = 0.

Signals that start at t = 0 are referred to as causal signals. The causal exponential

function is given by

e st t ≥ 0

x ( t ) = e st u ( t ) =

(1.41)

0 t < 0 ,

where we have used the unit step function to incorporate causality in the com-
plex exponential functions. Similarly, the causal implementation of the DT
exponential function is defined as follows:

e sk

k ≥ 0

x [ k ] = e sk u [ k ] =

(1.42)

0 k < 0 .

The same concept can be extended to derive causal implementations of sinu-
soidal and other non-causal signals.

Example 1.11
Plot the DT causal exponential function x [ k ] = e (j0 . 2 π - 0 . 05) k u [ k ] .

Solution
The real and imaginary components of the non-causal signal e (j0 . 2 π - 0 . 05) k are
plotted in Fig. 1.16. To plot its causal implementation, we multiply e (j0 . 2 π - 0 . 05) k
by the unit step function u [ k ]. This implies that the causal implementation will
be zero for k < 0. The real and imaginary components of the resulting function
are plotted in Fig. 1.17.