Now that we have the underpinnings of digital computation, we
need to return to signal processing ideas. The most prominent of which
is, of course, the Fourier transform. The Fourier transform of a
sequence is defined to be
Sei2πf=∑n=−∞∞sne−(i2πfn)
S
2f
n
sn
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)=
S
*
ej2πf
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
f1
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
f1
n
sn
2
f1
n
n
2
n
sn
2
f
n
n
sn
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
12
12
. When the
signal is real-valued, we can further simplify our plotting chores by
showing the spectrum only over
0
12
0
12
; 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
T
s
has a sampled waveform that equals
cos2π·×12Ts·nTs=cosπn=−1n
2
π
·
1
2
T
s
·
n
T
s
π
n
1
n
(3)
The exponential in the DTFT at frequency
12
1
2
equals
e−(i2πn)2=e−(iπn)=−1n
2
π
n
2
π
n
1
n
, meaning that the correspondence
between analog and discrete-time frequency is established:
f
D
=
f
A
Ts
f
D
f
A
T
s
(4)
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
Δ
decreases, the value of
c
0
c
0
, the largest Fourier coefficient, decreases to zero:
|
c
0
|=AΔT
c
0
A
Δ
T
. Thus, to maintain a mathematically viable Sampling Theorem, the amplitude
A
A
must increase as
1Δ
1
Δ
, becoming infinitely large as the pulse duration decreases. Practical systems use a small value of
Δ
Δ, say
0.1·Ts
0.1
·
T
s
and use amplifiers to rescale the signal. Thus, the sampled signal's spectrum becomes periodic with period
1Ts
1
T
s
. Thus, the Nyquist frequency
12Ts
1
2
T
s
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
π
f
n
∞
∞
a
n
u
n
2
π
f
n
n
∞
0
a
2
π
f
n
(5)
This sum is a special case of the geometric series.
∑n=0∞αn=11−α
,
|α|<1
n
0
∞
α
n
1
1
α
,
α
1
(6)
Thus, as long as
|a|<1
a
1
, we have our Fourier transform.
Sei2πf=11−ae−(i2πf)
S
2
π
f
1
1
a
2
π
f
(7)
Using Euler's relation, we can express the magnitude and phase of this spectrum.
|Sei2πf|=11−acos2πf2+a2sin22πf
S
2
π
f
1
1
a
2
π
f
2
a
2
2
π
f
2
(8)
∠Sei2πf=−tan-1asin2πf1−acos2πf
∠
S
2
π
f
a
2
π
f
1
a
2
π
f
(9)
No matter what value of
a
a
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
0
to
12
1
2
-- with increasing
a
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-
N
N
pulse sequence.
sn={1 if 0≤n≤N−10 otherwise
s
n
1
0
n
N
1
0
(10)
The Fourier transform of this sequence has the form of a truncated geometric series.
Sei2πf=∑n=0N−1e−(i2πfn)
S
2
π
f
n
0
N
1
2
π
f
n
(11)
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
(12)
for all values of
α
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.
α∑n=
n
0
N+
n
0
−1Did not convert apply/uplimitαn−∑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
(13)
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
π
f
1
2
π
f
N
1
2
π
f
π
f
N
1
π
f
N
π
f
(14)
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
π
f
π
f
N
1
dsinc
π
f
. The discrete-time pulse's spectrum contains many ripples, the number of which increase with
N
N, the pulse's duration.
The inverse discrete-time Fourier transform is easily derived from the following relationship:
∫−1212e−(i2πfm)e∗iπfndf={1 if m=n0 if m≠n
1
2
1
2
f
2
π
f
m
∗
π
f
n
1
m
n
0
m
n
(15)
Therefore, we find that
∫−1212Sei2πfe∗2πfndf=∫−1212∑msme−(i2πfm)e∗i2πfndf=∑msm∫−1212e(−(i2πf))(m−n)df=sn
f
1
2
1
2
S
2
π
f
∗
2
π
f
n
f
1
2
1
2
m
m
s
m
2
π
f
m
∗
2
π
f
n
m
m
s
m
f
1
2
1
2
2
π
f
m
n
s
n
(16)
The Fourier transform pairs in discrete-time are
Sei2πf=∑nsne−(i2πfn)
S
2
π
f
n
n
s
n
2
π
f
n
(17)
sn=∫−1212Sei2πfe∗i2πfndf
s
n
f
1
2
1
2
S
2
π
f
∗
2
π
f
n
(18)
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|2df
n
∞
∞
s
n
2
f
1
2
1
2
S
2
π
f
2
(19)
To show this important property, we simply substitute the Fourier transform expression into the frequency-domain expression for power.
∫−1212|Sei2πf|2df=∫−1212∑nsne−(i2πfn)∑m
s
∗
ne∗i2πfmdf=∑n,msn
s
∗
m∫−1212e∗i2πf(m−n)df
f
1
2
1
2
S
2
π
f
2
f
1
2
1
2
n
n
s
n
2
π
f
n
m
m
s
∗
n
∗
2
π
f
m
n,m
n,m
s
n
s
∗
m
f
1
2
1
2
∗
2
π
f
m
n
(20)
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
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
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
f
T
. Thus, the energy in the sampled signal equals the original signal's
energy multipled by T.
