Ch.4 Fourier Analysis of
Discrete-Time Signals
Kamen and Heck
4.1 Discrete-Time Fourier
Transform
• X() = n=-, x[n] e -jn (Eq. 4.1)
• Complex valued function of real variable
, the frequency.
• A sufficient condition for x[n] to have a
DTFT in the ordinary sense is that x[n] be
absolutely summable.
Example 4.1 Computation of the
DTFT
Consider x[n] = an, 0nq and 0 otherwise.
The DTFT is
X() = n=-, x[n] e -jn
= n=0,q an e -jn
= n=0,q (ae -j)n
= [1 – (ae -j)q+1 ] / [1- (ae -j)]
(where the closed form expression for a
partial sum exponential is used—(Eq.4.5)
4.1 Discrete-Time Fourier
Transform (cont.)
• X() is a periodic function of  with period
2.
• Rectangular Form: X() = R() + jI().
R() = n=-, x[n] cos(n)
I() = - n=-, x[n] sin(n)
• Polar Form: X() = |X()| +exp[j X()].
– |X()| = SQRT[R2() + I2()].
– X()=tan-1[I()/ R()] when R()  0
–
=  + tan-1[I()/ R()] when R() < 0
Example 4.2 Rectangular and Polar
Forms
– Consider x[n] = an u(n).
– This is similar to Ex. 4.1 except we have q.
– Consider the DTFT from Ex. 4.1 but let q:
• X() = lim q [1 – (ae -j)q+1 ] / [1- (ae -j)]
• This limit exists for |a| < 1.
• For this case, the DTFT exists in the ordinary
sense.
• X() = 1/ [1- (ae -j)] (Eq. 4.16)
• The rectangular and polar forms are shown on
pages 170-171.
4.1.1 Signals with Even or Odd
Symmetry
• Let x[n] be a real-valued discrete-time
signal that is an even function (ie, x[n] = x[n].)
– The DTFT is X()= x[0] + n=1, 2x[n] cos(n)
• Let x[n] be an odd function (ie,x[n]=-x[-n])
– The DTFT is X()= x[0] - n=1, j2x[n]sin(n)
Example 4.3 DTFT of Rectangular
Pulse
– Let p[n] = 1 for -q n  q and 0 elsewhere.
– The signal is even but it is easier to use 4.2.
– P() = n=-q,q e -jn
–
=[ e jq – e -j(q+1) ] / [1- e -j ]
–
= sin[(q + 1/2) ]/[sin(/2)]
– This is the discrete-time counterpart to the
transform of the rectangular pulse (Ex. 3.9).
– Figure 4.3 illustrates the DTFT.
4.1.2 Spectrum of a Discrete-Time
Signal
• For simplification, the discrete-time Fourier
series is not discussed.
• For a discrete time signal that is not a
function of sinusoids the spectrum is a
continuum of frequency components.
• The frequency spectrum is made up of the
amplitude spectrum and the phase
spectrum.
• The highest value of  = .
Example 4.4 Decaying Exponential
•
•
•
•
Assume that x[n] = (.5)n u(n).
The signal is plotted in Fig. 4.1.
The spectrum is shown in Figure 4.2
Note that most of the spectrum is in the
lower frequencies.
Example 4.5 Signal with HighFrequency Components
• Consider x[n] = (-.5)n u(n).
• From Figure 4.4 we see that there should
be higher frequency components in this
signal.
• From the result of Ex 4.2, the DTFT is:
– X() = 1/ [1- (-.5e -j)] = 1/ [1 + .5e -j]
– The amplitude and phase spectra are given
by equations 4.25 and 4.26 and plotted in
Figure 4.5.
4.1.3 Inverse DTFT
• x[n] = 1/2
2
0
X() e jn d (Eq. 4.7)
4.1.4 Generalized DTFT
• Example 4.6 DTFT of a Constant Signal
• Let x[n] =1 for all n.
• This signal does not have a DTFT in the
ordinary sense—(Why?)
• Figure 4.6 shows the generalized DTFT.
• Discussion on page 176 illustrates that its
inverse is the constant signal.
DTFT Transform Pairs and
Properties
• 4.1.5 Transform Pairs—Table 4.1 page
177.
• Properties—Table 4.2 page 178.
– No duality property, but there is a relationship
between the inverse of the CTFT and the
DTFT.
– Result can be used to generate DTFT pairs
from CTFT pairs—see Example 4.7.
4.2 Discrete Fourier Transform
• Let x[n] be a discrete-time signal.
• Let X() is the DTFT of x[n].
• Note: the DTFT is a continuous function
of .
• Let N be a positive integer, then the DFT
of x[n] is:
– Xk = n=0,N-1 x[n] e -j2kn/N , k=0,1,2,…N-1
4.2 The DFT (p.2)
• In general, Xk is a function of the discrete
integer k.
• There are N values in the DFT of x[n].
• These values are complex numbers.
• Polar form: Xk = |Xk| exp [jXk]
• Rectangular form: Xk = Rk + jIk
– See equations 4.36, 4.37.
• MATLAB—program on page 180.
4.2 The DFT (p.3)
• Example 4.8 Computation of the DFT
– Finite sequence –page 181.
• 4.2.1 Symmetry
– Magnitude of the DFT is symmetric about N/2, for N
even.
– Phase angle of the DFT has odd symmetry about N/2
when N is even.
• 4.2.2 Inverse DFT—see equation 4.40 and
MATLAB program and Example 4.9 on page
183.
The DFT (p.4)
• 4.2.3 Sinusoidal Form
– The right hand side of the IDFT equation can
be written as sinusoids.
– See equation 4.45 and Example 4.10.
• 4.2.4 Relationship to DTFT
– If x[n] = 0 for n<0 and n  N, the DFT Xk can
be viewed as a freqeuency sample version of
the DTFT.
– Xk =X() =2k/N = X(2k/N ), k = 0,1,2,…,N-1
Example 4.11 DTFT and DFT of a
Pulse
• Consider p[n] from example 4.3.
• Let x[n] be p[n-q].
• Figure 4.10 shows the amplitude spectrum
for q=5.
• Figure 4.11 shows the amplitude of the
DFT for q=5 and N= 22.
• Figure 4.12 shows the amplitude of the
DFT for q=5 and N = 88.
4.4 FFT Algorithm
• Consider the DFT and Inverse DFT:
• Xk = Σn=0,1,…,N-1 x[n] e -j2kn/N k=0,1,…,N-1
• x[n]= (1/N ) Σk=0,…,N-1 Xk e j2kn/N, n=0,…,N-1
• How many multiplications are needed to
compute the DFT? (N2)
• The FFT algorithm requires N(log2N)/2
multiplications.
4.4 FFT Algorithm (p.2)
• If N = 1024,
– DFT requires 1,048,576 multiplications
– FFT requires 5,120 multiplications
• There are different variations of the FFT
algorithm.
• One uses “decimation-in-time”.
4.4 FFT (p.3)
• Decimation-in-Time
– Subdivide the time interval into intervals
having a smaller number of points.
4.4 FFT (p.4)
• Xk can be broken up into two parts.
– First let exp(-j2/N) = WN
– Then Xk = Σn=0,1,…,N-1 x[n]( WN )kn k=0,1,…,N-1
– Let N be an even integer: a[n]=x[2n] ; b[n]=x[2n + 1],
for n = 0,…,N/2.
– Let Ak = Σ n=0,…,N/2-1 a[n] (WN/2)kn, k=0,1,…N/2-1
– Let Bk = Σ n=0,…,N/2-1 b[n] (WN/2)kn, k=0,1,…N/2-1
– Then Xk = Ak + (WN)k Bk, k=0,1,…,N/2 -1
– And X(N/2)+k = Ak - (WN)k Bk, k=0,1,…,N/2 -1
– See page 197 for the verification.
4.4 FFT (p.5)
• Note that the two parts are (N/2) DFTs.
• This can continue until signals with only
one nonzero value are obtained if N is a
power of 2.
• The process is graphically illustrated by
Figure 4.21.
• To have the outputs in the correct order, a
process called bit reversing (see Table
4.3) is used.
4.4.1 Applications of the FFT
Algorithm
• Computation of the Fourier Transform
• Convolution
• Data Analysis
– Extraction of a Sinusoidal Component Embedded in
Noise
– Analysis of Sunspot Data
– Stock Price Analysis