The Fourier
transform of the discrete-time signal
sn
s n
is defined to be
Sei2πf=∑
n
=−∞∞sne−(i2πfn)
S
2
f
n
s
n
2
f
n
(1)
Frequency here has no units. As should be expected, this
definition is linear, with the transform of a sum of signals
equaling the sum of their transforms. Real-valued signals have
conjugate-symmetric spectra:
Se−(i2πf)=Sej2πf¯
S
2
f
S
j
2
f
.
A special property of the discrete-time Fourier transform is
that it is periodic with period one:
Sei2π(f+1)=Sei2πf
S
2
f
1
S
2
f
.
Derive this property from the definition of the DTFT.
Sei2π(f+1)=∑
n
=−∞∞sne−(i2π(f+1)n)=∑
n
=−∞∞e−(i2πn)snei2πfn=∑
n
=−∞∞snei2πfn=Sei2πf
S
2
f
1
n
s
n
2
f
1
n
n
2
n
s
n
2
f
n
n
s
n
2
f
n
S
2
f
(2)
Because of this periodicity, we need only plot the spectrum over
one period to understand completely the spectrum's structure;
typically, we plot the spectrum over the frequency range
−12
12
1
2
1
2
.
When the signal is real-valued, we can further simplify our
plotting chores by showing the spectrum only over
0
12
0
1
2
;
the spectrum at negative frequencies can be derived from
positive-frequency spectral values.
When we obtain the discrete-time signal via sampling an analog
signal, the
Nyquist
frequency
corresponds to the discrete-time frequency
12
1
2
.
To show this, note that a sinusoid at the Nyquist frequency
12Ts
1
2
Ts
has a sampled waveform that equals
cos2π×12TsnTs=cosπn=−1n
2
1
2
T
s
n
T
s
n
1
n
The exponential in the DTFT at frequency
12
1
2
equals
e−i2πn2=e−(iπn)=−1n
2
n
2
n
1
n
,
meaning that the correspondence between analog and discrete-time
frequency is established:
f
D
=
f
A
Ts
f
D
f
A
Ts
(3)
where
f
D
f
D
and
f
A
f
A
represent discrete-time and analog frequency variables,
respectively. The
aliasing
figure
provides another way of deriving this result. As the duration
of each pulse in the periodic sampling signal
p
Ts
t
p
Ts
t
narrows, the amplitudes of the signal's spectral repetitions,
which are governed by the
Fourier series
coefficients
of
p
Ts
t
p
Ts
t
,
become increasingly equal. Examination of the
periodic pulse
signal
reveals that as Δ decreses, the value of
c
0
c
0
,
the largest Fourier coefficient, decreases to zero:
|
c
0
|=AΔTs
c
0
A
Δ
Ts
.
Thus, to maintain a mathematically viable Sampling Theorem, the
amplitude A must increase as
1Δ
1
Δ
,
becoming infinitely large as the pulse duration decreases.
Practical systems use a small value of Δ, say
0.1·
Ts
0.1
·
Ts
and use amplifiers to rescale the signal. Thus, the sampled
signal's spectrum becomes periodic with period
1Ts
1
Ts
.
Thus, the Nyquist frequency
12Ts
1
2
Ts
corresponds to the frequency
12
1
2
.
Let's compute the discrete-time Fourier transform of the
exponentially decaying sequence
sn=anun
s
n
a
n
u
n
,
where
un
u
n
is the unit-step sequence. Simply plugging the signal's
expression into the Fourier transform formula,
Sei2πf=∑
n
=−∞∞anune−(i2πfn)=∑
n
=0∞ae−(i2πf)n
S
2
f
n
a
n
u
n
2
f
n
n
0
a
2
f
n
(4)
This sum is a special case of the geometric
series.
∑
n
=0∞αn=∀
α
,|α|<1:11−α
n
0
α
n
α
α
1
1
1
α
(5)
Thus, as long as
|a|<1
a
1
,
we have our Fourier transform.
Sei2πf=11−ae−(i2πf)
S
2
f
1
1
a
2
f
(6)
Using Euler's relation, we can express the magnitude and phase
of this spectrum.
|Sei2πf|=11−acos2πf2+a2sin22πf
S
2
f
1
1
a
2
f
2
a
2
2
f
2
(7)
∠Sei2πf=−tan-1asin2πf1−acos2πf
∠
S
2
f
a
2
f
1
a
2
f
(8)
No matter what value of
aa we choose, the above formulae
clearly demonstrate the periodic nature of the spectra of
discrete-time signals.
Figure 1
shows indeed that the spectrum is a periodic function. We
need only consider the spectrum between
−12
1
2
and
12
1
2
to unambiguously define it. When
a>0
a
0
,
we have a lowpass spectrum—the spectrum diminishes as
frequency increases from 0 to
12
1
2
—with increasing a leading to a greater low frequency
content; for
a<0
a
0
,
we have a highpass spectrum
(Figure 2).
Analogous to the analog pulse signal, let's find the spectrum
of the length-NN pulse sequence.
sn={1 if 0≤n≤N−10 otherwise
s
n
1
0
n
N
1
0
(9)
The Fourier transform of this sequence has the form of a
truncated geometric series.
Sei2πf=∑
n
=0N−1e−(i2πfn)
S
2
f
n
0
N
1
2
f
n
(10)
For the so-called finite geometric series, we know that
∑
n
=
n
0
N+
n
0
−1αn=α
n
0
1−αN1−α
n
n
0
N
n
0
1
α
n
α
n
0
1
α
N
1
α
(11)
for all values of α.
Derive this formula for the finite geometric series sum.
The "trick" is to consider the difference between the
series' sum and the sum of the series multiplied by
α.
α∑n=
n
0
N+
n
0
−1αn−∑n=
n
0
N+
n
0
−1αn=αN+
n
0
−α
n
0
α
n
n
0
N
n
0
1
α
n
n
n
0
N
n
0
1
α
n
α
N
n
0
α
n
0
which, after manipulation, yields the geometric sum formula.
Applying this result yields
(Figure 3.)
Sei2πf=1−e−(i2πfN)1−e−(i2πf)=e−(iπf(N−1))sinπfNsinπf
S
2
f
1
2
f
N
1
2
f
f
N
1
f
N
f
(12)
The ratio of sine functions has the generic form of
sinNxsinx
N
x
x
,
which is known as the discrete-time sinc function
dsincx
dsinc
x
.
Thus, our transform can be concisely expressed as
Sei2πf=e−(iπf(N−1))dsincπf
S
2
f
f
N
1
dsinc
f
.
The discrete-time pulse's spectrum contains many ripples, the
number of which increase with N, the pulse's duration.
The inverse discrete-time Fourier transform is easily derived
from the following relationship:
∫−1212e−(i2πfm)ei2πfnd
f
={1 if m=n0 if m≠n
1
2
1
2
f
2
f
m
2
f
n
1
m
n
0
m
n
(13)
Therefore, we find that
∫−1212Sei2πfei2πfnd
f
=∫−1212∑msme−(i2πfm)ei2πfnd
f
=∑msm∫−1212e(−(i2πf))(m−n)d
f
=sn
f
1
2
1
2
S
2
f
2
f
n
f
1
2
1
2
m
m
s
m
2
f
m
2
f
n
m
m
s
m
f
1
2
1
2
2
f
m
n
s
n
(14)
The Fourier transform pairs in discrete-time are
Sei2πf=∑n=−∞∞sne−(i2πfn)
S
2
f
n
s
n
2
f
n
(15)
sn=∫−1212Sei2πfei2πfndf
s
n
f
1
2
1
2
S
2
f
2
f
n
(16)
The properties of the discrete-time Fourier transform mirror
those of the analog Fourier transform. The
DTFT properties table
shows similarities and differences. One important common
property is Parseval's Theorem.
∑
n
=−∞∞|sn|2=∫−1212|Sei2πf|2d
f
n
s
n
2
f
1
2
1
2
S
2
f
2
(17)
To show this important property, we simply substitute the
Fourier transform expression into the frequency-domain
expression for power.
∫−1212|Sei2πf|2d
f
=∫−1212∑nsne−(i2πfn)∑msn¯ei2πfmd
f
=∑n,msnsn¯∫−1212ei2πf(m−n)d
f
f
1
2
1
2
S
2
f
2
f
1
2
1
2
n
n
s
n
2
f
n
m
m
s
n
2
f
m
,
n
m
,
n
m
s
n
s
n
f
1
2
1
2
2
f
m
n
(18)
Using the
orthogonality
relation,
the integral equals
δm−n
δ
m
n
,
where
δn
δ
n
is the
unit
sample.
Thus, the double sum collapses into a single sum because nonzero
values occur only when n = m, giving Parseval's Theorem as a
result. We term
∑ns2n
n
n
s
n
2
the energy in the discrete-time signal
sn
s
n
in spite of the fact that discrete-time signals don't consume
(or produce for that matter) energy. This terminology is a
carry-over from the analog world.
Suppose we obtained our discrete-time signal from values of
the product
st
p
T
s
t
s
t
p
T
s
t
,
where the duration of the component pulses in
p
T
s
t
p
T
s
t
is Δ. How is the discrete-time signal energy related
to the total energy contained in
st
s
t
?
Assume the signal is bandlimited and that the sampling rate
was chosen appropriate to the Sampling Theorem's conditions.
If the sampling frequency exceeds the Nyquist frequency, the
spectrum of the samples equals the analog spectrum, but over
the normalized analog frequency fT. Thus, the energy in the
sampled signal equals the original signal's energy multipled
by T.
