Discrete-time signals and systems
See Oppenheim and Schafer, Second Edition pages 8–93, or First Edition
pages 8–79.
1 Discrete-time signals
A discrete-time signal is represented as a sequence of numbers:
x D fxŒng;
1 < n < 1:
Here n is an integer, and xŒn is the nth sample in the sequence.
Discrete-time signals are often obtained by sampling continuous-time signals.
In this case the nth sample of the sequence is equal to the value of the analogue
signal xa .t / at time t D nT :
xŒn D xa .nT /;
1 < n < 1:
The sampling period is then equal to T , and the sampling frequency is
fs D 1=T .
xa .1T /
...
...
t
x[1]
For this reason, although xŒn is strictly the nth number in the sequence, we
often refer to it as the nth sample. We also often refer to “the sequence xŒn”
when we mean the entire sequence.
Discrete-time signals are often depicted graphically as follows:
1
x[−3]
x[−2]
x[−4]
x[−1]
x[4]
x[0]
1
...
−4
−3
−2
...
2
3
−1 0
4
x[1]
n
x[3]
x[2]
(This can be plotted using the MATLAB function stem.) The value xŒn is
undefined for noninteger values of n.
Sequences can be manipulated in several ways. The sum and product of two
sequences xŒn and yŒn are defined as the sample-by-sample sum and product
respectively. Multiplication of xŒn by a is defined as the multiplication of
each sample value by a.
A sequence yŒn is a delayed or shifted version of xŒn if
yŒn D xŒn
n0 ;
with n0 an integer.
The unit sample sequence
1
n
0
is defined as
ıŒn D
8
<0
n¤0
:1
n D 0:
This sequence is often referred to as a discrete-time impulse, or just impulse.
It plays the same role for discrete-time signals as the Dirac delta function does
for continuous-time signals. However, there are no mathematical
2
complications in its definition.
An important aspect of the impulse sequence is that an arbitrary sequence can
be represented as a sum of scaled, delayed impulses. For example, the
sequence
a
a
3
a
4
2
a
1
a4
a0
...
...
1
−4
−3
−2
−1 0
2
3
n
4
a1
a3
a2
can be represented as
xŒn D a
4 ıŒn
C 4 C a
3 ıŒn
C 3 C a
Ca1 ıŒn
2 ıŒn
1 C a2 ıŒn
C 2 C a
2 C a3 ıŒn
1 ıŒn
C 1 C a0 ıŒn
3 C a4 ıŒn
In general, any sequence can be expressed as
xŒn D
1
X
xŒkıŒn
k:
kD 1
The unit step sequence
1
n
0
is defined as
uŒn D
8
<1
n0
:0
n < 0:
3
4:
The unit step is related to the impulse by
n
X
uŒn D
ıŒk:
kD 1
Alternatively, this can be expressed as
uŒn D ıŒn C ıŒn
1 C ıŒn
2 C D
1
X
ıŒn
k:
kD0
Conversely, the unit sample sequence can be expressed as the first backward
difference of the unit step sequence
ıŒn D uŒn
uŒn
1:
Exponential sequences are important for analysing and representing
discrete-time systems. The general form is
xŒn D A˛ n :
If A and ˛ are real numbers then the sequence is real. If 0 < ˛ < 1 and A is
positive, then the sequence values are positive and decrease with increasing n:
...
...
n
0
For 1 < ˛ < 0 the sequence alternates in sign, but decreases in magnitude.
For j˛j > 1 the sequence grows in magnitude as n increases.
A sinusoidal sequence
...
0
4
n
...
has the form
xŒn D A cos.!0 n C /
for all n;
with A and real constants. The exponential sequence A˛ n with complex
˛ D j˛je j!0 and A D jAje j can be expressed as
xŒn D A˛ n D jAje j j˛jn e j!0 n D jAjj˛jn e j.!0 nC/
D jAjj˛jn cos.!0 n C / C j jAjj˛jn sin.!0 n C /;
so the real and imaginary parts are exponentially weighted sinusoids.
When j˛j D 1 the sequence is called the complex exponential sequence:
xŒn D jAje j.!0 nC/ D jAj cos.!0 n C / C j jAj sin.!0 n C /:
The frequency of this complex sinusoid is !0 , and is measured in radians per
sample. The phase of the signal is .
The index n is always an integer. This leads to some important differences
between the properties of discrete-time and continuous-time complex
exponentials:
Consider the complex exponential with frequency .!0 C 2/:
xŒn D Ae j.!0 C2/n D Ae j!0 n e j 2 n D Ae j!0 n :
Thus the sequence for the complex exponential with frequency !0 is
exactly the same as that for the complex exponential with frequency
.!0 C 2/. More generally, complex exponential sequences with
frequencies .!0 C 2 r/, where r is an integer, are indistinguishable from
one another. Similarly, for sinusoidal sequences
xŒn D A cosŒ.!0 C 2 r/n C  D A cos.!0 n C /:
In the continuous-time case, sinusoidal and complex exponential
sequences are always periodic. Discrete-time sequences are periodic (with
period N) if
xŒn D xŒn C N 
for all n:
5
Thus the discrete-time sinusoid is only periodic if
A cos.!0 n C / D A cos.!0 n C !0 N C /;
which requires that
!0 N D 2k
for k an integer:
The same condition is required for the complex exponential sequence
C e j!0 n to be periodic.
The two factors just described can be combined to reach the conclusion that
there are only N distinguishable frequencies for which the corresponding
sequences are periodic with period N . One such set is
!k D
2k
;
N
k D 0; 1; : : : ; N
1:
Additionally, for discrete-time sequences the interpretation of high and low
frequencies has to be modified: the discrete-time sinusoidal sequence
xŒn D A cos.!0 n C / oscillates more rapidly as !0 increases from 0 to ,
but the oscillations become slower as it increases further from to 2.
6
!0 D 0
1
0
−1
!0 D 8
1
0
−1
!0 D 4
1
0
−1
!0 D 1
0
!0 D 15
8
!0 D 7
4
−1
1
0
−1
1
0
−1
−8
−6
−4
−2
0
2
4
6
8
The sequence corresponding to !0 D 0 is indistinguishable from that with
!0 D 2. In general, any frequencies in the vicinity of !0 D 2k for integer
k are typically referred to as low frequencies, and those in the vicinity of
!0 D . C 2k/ are high frequencies.
7
2 Discrete-time systems
A discrete-time system is defined as a transformation or mapping operator that
maps an input signal xŒn to an output signal yŒn. This can be denoted as
yŒn D T fxŒng:
T fg
x[n]
y[n]
Example: Ideal delay
yŒn D xŒn
nd  W
x[n]
...
...
n
−3
−2
−1
0
1
2
3
y[n]=x[n−2]
...
...
n
−1
0
1
2
3
This operation shifts input sequence later by nd samples.
8
4
5
Example: Moving average
M2
X
1
xŒn
yŒn D
M1 C M2 C 1
k
kD M1
For M1 D 1 and M2 D 1, the input sequence
y[3]
x[n]
...
...
n
−1
0
1
2
3
4
5
y[2]
yields an output with
::
:
1
.xŒ1 C xŒ2 C xŒ3/
3
1
yŒ3 D .xŒ2 C xŒ3 C xŒ4/
3
::
:
yŒ2 D
In general, systems can be classified by placing constraints on the
transformation T fg.
2.1 Memoryless systems
A system is memoryless if the output yŒn depends only on xŒn at the same n.
For example, yŒn D .xŒn/2 is memoryless, but the ideal delay
9
yŒn D xŒn
nd  is not unless nd D 0.
2.2 Linear systems
A system is linear if the principle of superposition applies. Thus if y1 Œn is the
response of the system to the input x1 Œn, and y2 Œn the response to x2 Œn, then
linearity implies
Additivity:
T fx1 Œn C x2 Œng D T fx1 Œng C T fx2 Œng D y1 Œn C y2 Œn
Scaling:
T fax1 Œng D aT fx1 Œng D ay1 Œn:
These properties combine to form the general principle of superposition
T fax1 Œn C bx2 Œng D aT fx1 Œng C bT fx2 Œng D ay1 Œn C by2 Œn:
In all cases a and b are arbitrary constants.
This property generalises to many inputs, so the response of a linear system to
P
P
xŒn D k ak xk Œn will be yŒn D k ak yk Œn.
2.3 Time-invariant systems
A system is time invariant if a time shift or delay of the input sequence causes
a corresponding shift in the output sequence. That is, if yŒn is the response to
xŒn, then yŒn n0  is the response to xŒn n0 .
For example, the accumulator system
yŒn D
n
X
kD 1
10
xŒk
is time invariant, but the compressor system
yŒn D xŒM n
for M a positive integer (which selects every M th sample from a sequence) is
not.
2.4 Causality
A system is causal if the output at n depends only on the input at n and earlier
inputs.
For example, the backward difference system
yŒn D xŒn
xŒn
1
is causal, but the forward difference system
yŒn D xŒn C 1
xŒn
is not.
2.5 Stability
A system is stable if every bounded input sequence produces a bounded output
sequence:
Bounded input: jxŒnj Bx < 1
Bounded output: jyŒnj By < 1.
For example, the accumulator
yŒn D
n
X
kD 1
11
xŒn
is an example of an unbounded system, since its response to the unit step uŒn
is
8
n
<0
X
n<0
yŒn D
uŒn D
:n C 1
n 0;
kD 1
which has no finite upper bound.
3 Linear time-invariant systems
If the linearity property is combined with the representation of a general
sequence as a linear combination of delayed impulses, then it follows that a
linear time-invariant (LTI) system can be completely characterised by its
impulse response.
Suppose hk Œn is the response of a linear system to the impulse ıŒn
n D k. Since
( 1
)
X
yŒn D T
xŒkıŒn k ;
k at
kD 1
the principle of superposition means that
yŒn D
1
X
xŒkT fıŒn
kg D
kD 1
1
X
xŒkhk Œn:
kD 1
If the system is additionally time invariant, then the response to ıŒn
hŒn k. The previous equation then becomes
yŒn D
1
X
xŒkhŒn
k is
k:
kD 1
This expression is called the convolution sum. Therefore, a LTI system has
the property that given hŒn, we can find yŒn for any input xŒn. Alternatively,
yŒn is the convolution of xŒn with hŒn, denoted as follows:
yŒn D xŒn hŒn:
12
The previous derivation suggests the interpretation that the input sample at
n D k, represented by xŒkıŒn k, is transformed by the system into an
output sequence xŒkhŒn k. For each k, these sequences are superimposed
to yield the overall output sequence:
hŒn
xŒn
1
−1 0
n
0
xŒ 1hŒn C 1
xŒ 1ıŒn C 1
−1 0
n
xŒ1ıŒn
n
0
n
xŒ1hŒn
1
1
1
0
n
0
yŒn D xŒ 1hŒn C 1 C xŒ1hŒn
0
n
1
n
A slightly different interpretation, however, leads to a convenient
computational form: the nth value of the output, namely yŒn, is obtained by
multiplying the input sequence (expressed as a function of k) by the sequence
with values hŒn k, and then summing all the values of the products
xŒkhŒn k. The key to this method is in understanding how to form the
sequence hŒn k for all values of n of interest.
To this end, note that hŒn k D hŒ .k n/. The sequence hŒ k is seen to
be equivalent to the sequence hŒk reflected around the origin:
13
h[k]
−2
k
0
5
Reflect
h[−k]
0
−5
k
2
Shift
h[n−k]
n−5
The sequence hŒn
to k D n.
0
n
n+2
k
k is then obtained by shifting the origin of the sequence
To implement discrete-time convolution, the sequences xŒk and hŒn k are
multiplied together for 1 < k < 1, and the products summed to obtain the
value of the output sample yŒn. To obtain another output sample, the
procedure is repeated with the origin shifted to the new sample position.
Example: analytical evaluation of the convolution sum
Consider the output of a system with impulse response
8
<1
0nN 1
hŒn D
:0
otherwise
to the input xŒn D an uŒn. To find the output at n, we must form the sum over
all k of the product xŒkhŒn k.
14
x[n]
1
0.5
0
−10
−5
0
k
−5
n−(N−1) 0
k
5
10
5
10
h[n−k]
1
0.5
0
−10
n
Since the sequences are non-overlapping for all negative n, the output must be
zero:
yŒn D 0;
For 0 n N
follows that
n < 0:
1 the product terms in the sum are xŒkhŒn
yŒn D
n
X
ak ;
0nN
k D ak , so it
1:
kD0
Finally, for n > N 1 the product terms are xŒkhŒn k D ak as before, but
the lower limit on the sum is now n N C 1. Therefore
yŒn D
n
X
ak ;
kDn N C1
15
n>N
1:
4 Properties of LTI systems
All LTI systems are described by the convolution sum
yŒn D
1
X
xŒkhŒn
k:
kD 1
Some properties of LTI systems can therefore be found by considering the
properties of the convolution operation:
Commutative: xŒn hŒn D hŒn xŒn
Distributive over addition:
xŒn .h1 Œn C h2 Œn/ D xŒn h1 Œn C xŒn h2 Œn:
Cascade connection:
x[n]
x[n]
h1 Œn
h2 Œn
h2 Œn
y[n]
h1 Œn
y[n]
yŒn D hŒn xŒn D h1 Œn h2 Œn xŒn D h2 Œn h1 Œn xŒn.
Parallel connection:
h1 Œn
x[n]
y[n]
h2 Œn
yŒn D .h1 Œn C h2 Œn/ xŒn D hp Œn xŒn.
Additional important properties are:
A LTI system is stable if and only if S D
16
P1
kD 1
jhŒkj < 1. The ideal
delay system hŒn D ıŒn
average system
nd  is stable since S D 1 < 1; the moving
M2
X
1
hŒn D
ıŒn k
M1 C M2 C 1
kD M1
8
1
<
M1 n M2
M1 CM2 C1
D
:0
otherwise;
the forward difference system hŒn D ıŒn C 1 ıŒn, and the backward
difference system hŒn D ıŒn ıŒn 1 are stable since S is the sum of a
finite number of finite samples, and is therefore less than 1; the
accumulator system
hŒn D
n
X
ıŒk
kD 1
D
8
<1
:0
n0
n<0
D uŒn
is unstable since S D
P1
nD0
uŒn D 1.
A LTI system is causal if and only if hŒn D 0 for n < 0. The ideal delay
system is causal if nd 0; the moving average system is causal if
M1 0 and M2 0; the accumulator and backward difference systems
are causal; the forward difference system is noncausal.
Systems with only a finite number of nonzero values in hŒn are called Finite
duration impulse response (FIR) systems. FIR systems are stable if each
impulse response value is finite. The ideal delay, the moving average, and the
forward and backward difference described above fall into this class. Infinite
impulse response (IIR) systems, such as the accumulator system, are more
difficult to analyse. For example, the accumulator system is unstable, but the
17
IIR system
hŒn D an uŒn;
jaj < 1
is stable since
SD
1
X
n
ja j 1
X
jajn D
nD0
nD0
1
1
jaj
<1
(it is the sum of an infinite geometric series).
Consider the system
Forward
difference
One−sample
delay
which has
hŒn D .ıŒn C 1
ıŒn/ ıŒn
D ıŒn
1 ıŒn C 1
D ıŒn
ıŒn
ıŒn
1
1 ıŒn
1:
This is the impulse response of a backward difference system:
Forward
difference
One−sample
delay
Backward
difference
Here a non-causal system has been converted to a causal one by cascading with
a delay. In general, any non-causal FIR system can be made causal by
cascading with a sufficiently long delay.
Consider the system consisting of an accumulator followed by a backward
difference:
18
Backward
difference
Accumulator
The impulse response of this system is
hŒn D uŒn .ıŒn
ıŒn
1/ D uŒn
uŒn
1 D ıŒn:
The output is therefore equal to the input because xŒn ıŒn D xŒn. Thus the
backward difference exactly compensates for (or inverts) the effect of the
accumulator — the backward difference system is the inverse system for the
accumulator, and vice versa. We define this inverse relationship for all LTI
systems:
hŒn hi Œn D ıŒn:
5 Linear constant coefficient difference equations
Some LTI systems can be represented in terms of linear constant coefficient
difference (LCCD) equations
N
X
ak yŒn
k D
M
X
bm xŒn
m:
mD0
kD0
Example: difference equation representation of the accumulator
Take for example the accumulator
Accumulator
x[n]
y[n]
Backward
difference
x[n]
Here yŒn yŒn 1 D xŒn, which can be written in the desired form with
N D 1, a0 D 1, a1 D 1, M D 0, and b0 D 1. Rewriting as
yŒn D yŒn
19
1 C xŒn
we obtain the recursion representation
x[n]
y[n]
One−sample
delay
where at n we add the current input xŒn to the previously accumulated sum
yŒn 1.
Example: difference equation representation of moving average
Consider now the moving average system with M1 D 0:
hŒn D
1
.uŒn
M2 C 1
uŒn
M2
1/:
The output of the system is
M2
X
1
yŒn D
xŒn
M2 C 1
k;
kD0
which is a LCCDE with N D 0, a0 D 1, and M D M2 , bk D 1=.M2 C 1/.
Using the sifting property of ıŒn,
hŒn D
1
.ıŒn
M2 C 1
ıŒn
M2
1/ uŒn
so
Attenuator
x[n]
+
x1 Œn
1=.M2 C 1/
.M2 C 1/
sample delay
20
−
Accumulator
y[n]
Here x1 Œn D 1=.M2 C 1/.xŒn xŒn
yŒn yŒn 1 D x1 Œn. Therefore
yŒn
yŒn
1 D
M2
1
.xŒn
M2 C 1
1/ and for the accumulator
xŒn
M2
1/;
which is again a (different) LCCD equation with N D 1, a0 D 1, a1 D
b0 D bM2 C1 D 1=.M2 C 1/.
1,
As for constant coefficient differential equations in the continuous case,
without additional information or constraints a LCCDE does not provide a
unique solution for the output given an input. Specifically, suppose we have
the particular output yp Œn for the input xp Œn. The same equation then has the
solution
yŒn D yp Œn C yh Œn;
where yh Œn is any solution with xŒn D 0. That is, yh Œn is an homogeneous
solution to the homogeneous equation
N
X
ak yh Œn
k D 0:
kD0
It can be shown that there are N nonzero solutions to this equation, so a set of
N auxiliary conditions are required for a unique specification of yŒn for a
given xŒn.
If a system is LTI and causal, then the initial conditions are initial rest
conditions, and a unique solution can be obtained.
6 Frequency-domain representation of
discrete-time signals and systems
The Fourier transform considered here is strictly speaking the discrete-time
Fourier transform (DTFT), although Oppenheim and Schafer call it just the
21
Fourier transform. Its properties are recapped here (with examples) to show
nomenclature.
Complex exponentials
xŒn D e j!n ;
1<n<1
are eigenfunctions of LTI systems:
yŒn D
1
X
hŒke j!.n
k/
1
X
D e j!n
kD 1
hŒke
j!k
kD 1
!
:
Defining
H.e
j!
/D
1
X
j!k
hŒke
kD 1
we have that yŒn D H.e j! /e j!n D H.e j! /xŒn. Therefore, e j!n is an
eigenfunction of the system, and H.e j! / is the associated eigenvalue.
The quantity H.e j! / is called the frequency response of the system, and
H.e j! / D HR .e j! / C jHI .e j! / D jH.e j! /je j ^H.e
j! /
:
Example: frequency response of ideal delay:
Consider the input xŒn D e j!n to the ideal delay system yŒn D xŒn
the output is
yŒn D e j!.n nd / D e j!nd e j!n :
The frequency response is therefore
H.e j! / D e
j!nd
Alternatively, for the ideal delay hŒn D ıŒn
H.e
j!
/D
1
X
ıŒn
nd e
:
nd ,
j!n
De
j!nd
nD 1
The real and imaginary parts of the frequency response are
22
:
nd :
HR .e j! / D cos.!nd / and HI .e j! / D sin.!nd /, or alternatively
jH.e j! /j D 1
^H.e j! / D
!nd :
The frequency response of a LTI system is essentially the same for continuous
and discrete time systems. However, an important distinction is that in the
discrete case it is always periodic in frequency with a period 2:
H.e
j.!C2/
/D
D
D
1
X
nD 1
1
X
nD 1
1
X
hŒne
j.!C2/n
hŒne
j!n
hŒne
j!n
e
j 2 n
D H.e j! /:
nD 1
This last result holds since e ˙j 2 n D 1 for integer n.
The reason for this periodicity is related to the observation that the sequence
˚ j!n
;
1<n<1
e
has exactly the same values as the sequence
o
n
j.!C2/n
;
1 < n < 1:
e
A system will therefore respond in exactly the same way to both sequences.
Example: ideal frequency selective filters
The frequency response of an ideal lowpass filter is as follows:
23
Hlp .e j! /
1
2
!c
0
!c
!
2
Only required part
Due to the periodicity in the response, it is only necessary to consider one
frequency cycle, usually chosen to be the range to . Other examples of
ideal filters are:
Hhp .e j! /
1
Highpass
0
!c
!c
!
j!
1 Hbs .e /
Bandstop
!b
0
!a
1
!a
Hbp .e j! /
!b
!
Bandpass
!b
!a
0
!a
!b
!
In these cases it is implied that the frequency response repeats with period 2
outside of the plotted interval.
Example: frequency response of the moving-average system
24
The frequency response of the moving average system
8
1
<
M1 n M2
M1 CM2 C1
hŒn D
:0
otherwise
is given by
H.e j! / D
1
e j!.M2 CM1 C1/=2 e j!.M2 CM1 C1/=2
e
M1 C M2 C 1
1 e j!
e j!.M2 CM1 C1/=2 e j!.M2 CM1 C1/=2
1
e
D
M1 C M2 C 1
e j!=2 e j!=2
sinŒ!.M1 C M2 C 1/=2 j!.M2 M1 /
1
2
e
:
D
M1 C M2 C 1
sin.!=2/
j!.M2
2
M1 C1/
j!.M2 M1 /
2
For M1 D 0 and M2 D 4,
jH.e j! /j
1
0
2
2
5
0
2
5
2
2
^H.e j! /
!
0
2
0
!
This system attenuates high frequencies (at around ! D ), and therefore has
the behaviour of a lowpass filter.
25
7 Fourier transforms of discrete sequences
The discrete time Fourier transform (DTFT) of the sequence xŒn is
X.e
j!
/D
1
X
xŒne
j!n
:
nD 1
This is also called the forward transform or analysis equation. The inverse
Fourier transform, or synthesis formula, is given by the Fourier integral
Z 1
xŒn D
X.e j! /e j!n d!:
2 The Fourier transform is generally a complex-valued function of !:
X.e j! / D XR .e j! / C jXI .e j! / D jX.e j! /je j ^X.e
j! /
:
The quantities jX.e j! /j and ^X.e j! / are referred to as the magnitude and
phase of the Fourier transform. The Fourier transform is often referred to as
the Fourier spectrum.
Since the frequency response of a LTI system is given by
H.e
j!
/D
1
X
hŒke
j!k
;
kD 1
it is clear that the frequency response is equivalent to the Fourier transform of
the impulse response, and the impulse response is
Z 1
hŒn D
H.e j! /e j!n d!:
2 A sufficient condition for the existence of the Fourier transform of a sequence
P
xŒn is that it be absolutely summable: 1
nD 1 jxŒnj < 1. In other words,
P1
the Fourier transform exists if the sum nD 1 jxŒnj converges. The Fourier
transform may however exist for sequences where this is not true — a rigorous
mathematical treatment can be found in the theory of generalised functions.
26
8 Symmetry properties of the Fourier transform
Any sequence xŒn can be expressed as
xŒn D xe Œn C xo Œn;
where xe Œn is conjugate symmetric (xe Œn D xe Œ n) and xo Œn is conjugate
antisymmetric (xo Œn D xo Œ n). These two components of the sequence
can be obtained as:
1
.xŒn C x Œ n/ D xe Œ n
2
1
xo Œn D .xŒn x Œ n/ D xo Œ n:
2
xe Œn D
If a real sequence is conjugate symmetric, then it is an even sequence, and if
conjugate antisymmetric, then it is odd.
Similarly, the Fourier transform X.e j! / can be decomposed into a sum of
conjugate symmetric and antisymmetric parts:
X.e j! / D Xe .e j! / C Xo .e j! /;
where
1
ŒX.e j! / C X .e
2
1
Xo .e j! / D ŒX.e j! / X .e
2
Xe .e j! / D
j!
j!
/
/:
With these definitions, and letting
X.e j! / D XR .e j! / C jXI .e j! /;
the symmetry properties of the Fourier transform can be summarised as
follows:
27
Sequence xŒn
Transform X.e j! /
x Œn
X .e
j!
/
x Œ n
X .e j! /
RefxŒng
Xe .e j! /
j ImfxŒng
Xo .e j! /
xe Œn
XR .e j! /
xo Œn
jXI .e j! /
Most of these properties can be proved by substituting into the expression for
the Fourier transform. Additionally, for real xŒn the following also hold:
Transform X.e j! /
Real sequence xŒn
xŒn
X.e j! / D X .e
xŒn
XR .e j! / D XR .e
xŒn
XI .e j! / D
xŒn
jX.e j! /j D jX.e
xŒn
^X.e j! / D
j!
j!
j!
^X.e
XR .e j! /
xo Œn
jXI .e j! /
/
j!
XI .e
xe Œn
/
/
/j
j!
/
9 Fourier transform theorems
Let X.e j! / be the Fourier transform of xŒn. The following theorems then
apply:
28
Sequences xŒn, yŒn
Transforms X.e j! /, Y .e j! /
Property
axŒn C byŒn
aX.e j! / C bY .e j! /
Linearity
xŒn
nd 
e
j!nd
X.e j! /
e j!0 n xŒn
X.e j.!
xŒ n
X.e
nxŒn
j
xŒn yŒn
X.e
1
2
xŒnyŒn
R
!0 /
j!
Time shift
/
Frequency shift
/
Time reversal
dX.e j! /
Frequency diff.
d!
j!
/Y .e
j!
X.e j /Y .e j.!
/
Convolution
/
/d
Modulation
Some useful Fourier transform pairs are:
Sequence
Fourier transform
ıŒn
1
ıŒn
n0 
1 . 1 < n < 1/
an uŒn .jaj < 1/
uŒn
.n C 1/an uŒn .jaj < 1/
sin.!c n/
n
xŒn D
8
<1
:0
e
j!n0
P1
kD 1 2ı.! C 2k/
1
1 ae j!
P1
1
C
kD 1 ı.! C 2k/
1 e j!
1
8.1 ae j! /2
X.e
j!
/D
<1
:0
0nM
j!j < !c
!c < j!j sinŒ!.M C1/=2
e j!M=2
sin.!=2/
otherwise
P1
e j!0 n
kD 1
29
2ı.!
!0 C 2k/