Signal representation using Fourier series - Continuous and Discrete Time Signals and Systems - page 150

complex exponential with the same frequency, except for possible changes in
its amplitude and phase.

Theorem 4.1 If a complex exponential function is applied to an LTIC system
with a real-valued impulse response function, the output response of the system
is identical to the complex exponential function except for changes in amplitude
and phase. In other words,
k 1 e j ω 1 t A 1 k 1 e j( ω 1 t +φ 1 ) , →

where A 1 and φ 1 are constants.

Proof
Assume that the complex exponential function x ( t ) = k 1 exp( j ω 1 t ) is applied to
an LTIC system with impulse response h ( t ). The output of the system is given
by the convolution of the input signal x ( t ) and the impulse response h ( t )is
given by

∞ ∞

k 1 e j ω 1 t

− j ω 1 τ d τ.

y ( t ) = h ( τ ) x ( t − τ )d τ = h ( τ )e (4.21)

−∞ −∞

Defining

∞

− j ωτ d τ,

H ( ω ) = h ( τ )e (4.22)

−∞

Eq. (4.21) can be expressed as follows:

y ( t ) = k 1 e j ω 1 t H ( ω 1 ) . (4.23)
From the definition in Eq. (4.22), we observe that H ( ω 1 ) is a complex-valued
constant, for a given value of ω 1 , such that it can be expressed as H ( ω 1 ) =

A 1 exp( j φ 1 ). In other words, A 1 is the magnitude of the complex constant H ( ω 1 )
and φ 1 is the phase of H ( ω 1 ). Expressing H ( ω 1 ) =

A 1 exp( j φ 1 ) in Eq. (4.23),

we obtain

A 1 k 1 e j( ω 1 t +φ 1 ) , y ( t ) =

which proves Theorem 4.1.

Corollary 4.1 The output response of an LTIC system, characterized by a real-
valued impulse response h ( t ), to a sinusoidal input is another sinusoidal function
with the same frequency, except for possible changes in its amplitude and phase.
In other words,

k 1 sin( ω 1 t ) → A 1 k 1 sin( ω 1 t + φ 1 ) (4.24)

and

k 1 cos( ω 1 t ) → A 1 k 1 cos( ω 1 t + φ 1 ) , (4.25)
where constants A 1 and φ 1 are the magnitude and phase of H ( ω 1 ) defined in
Eq. (4.22) with ω set to ω 1 .