Signals and Systems - Electrical Engineering

10.2 Discrete-Time Fourier Transform 591

In particular, ifφ=0, the signalx[n] is a cosine and its phase is zero. Ifφ=−π/2,x[n] is a sine

and its phase isπ/2 atω=−ω 0 and−π/2 atω=−ω 0. The DTFT of a sine is

X(ejω)=π

[

δ(ω−ω 0 )e−jπ/^2 +δ(ω+ω 0 )ejπ/^2

]

The DTFTs of the cosine and the sine are only different in the phase. n

The symmetry property, like other properties, also applies to systems. Ifh[n]is the impulse response of an LTI

discrete-time system, and it is real valued, its DTFT is

H(ejω)=Z

(

h[n]

)∣∣

z=ejω=H(z)

∣

∣z=ejω

if the region of convergence ofH(z)includes the unit circle. As with the DTFT of a signal, the frequency

response of the system,H(ejω), has a magnitude that is an even function ofω, and a phase that is an odd

function ofω. Thus, themagnitude responseof the system is such that

|H(ejω)|=|H(e−jω)| (10.23)

and thephase responseis such that

∠H(ejω)=−∠H(e−jω) (10.24)

According to these symmetries and that the frequency response is periodic, it is only necessary to give these

responses in[0,π]rather than in(−π,π].

Computation of the Phase Spectrum
Computation of the phase using MATLAB is complicated by the following three issues:

n Definition of the phase of a complex number: Given a complex numberz=x+jy=|z|ejθ, its phase
θis computed using the inverse tangent function

θ=tan−^1

(y

x

)

This computation is not well defined because the principal values of tan−^1 are [−π/2,π/2], while
the phase can extend beyond those values. By adding the information of which quadrantsxandy
are in, the principal values can be extended to [−π,π). When the phase is linear (i.e.,θ=−Nω
for some integerN), using the extended principal values is not good enough.
n Significance of magnitude when computing phase: Given two complex numbers,z 1 = 1 +j=

√

2 ejπ/^4

andz 2 =z 1 × 10 −^16 =

√

2 × 10 −^16 ejπ/^4 , they both have the same phase ofπ/4 but the magni-
tudes are very different,|z 2 |= 10 −^16 |z 1 |. For practical purposes, the magnitude ofz 1 is more
significant than that ofz 2 , which is very close to zero, so one could disregard the phase ofz 2 with
no effect on computations.
n Noisy measurements: Given that noise is ever present in actual measurements, even very small noise
present in the signal can change the computation of phase.