Introduction & Review
Sk. Mohammadul Haque
DST INSPIRE Faculty
Department of Electrical Engineering
Indian Institute of Technology Kharagpur
Digital Signal Processing 2023a
Sk. Mohammadul Haque
Introduction & Review
Digital Signal Processing (Part One)
Reference materials:
Text book:
Digital Signal Processing: John G. Proakis and Dimitris G.
Manolakis
Reference books:
Discrete-Time Signal Processing – Alan V. Oppenheim and
Ronald W. Schafe
Other articles:
As will be discussed
Sk. Mohammadul Haque
Introduction & Review
Signals
Signals
are mathematical functions of a number of independent
variables that are usually called indices.
100πn
e.g. f (t) = 100 sin(100πt), f [n] = 100 sin(
),
T
m
n
e.g. f (x , y ) = 2x + y , f [m.n] = 2
+
T1
T2
voltage signals, current signals, position, velocity, momentum,
volume, flow rate, etc.
multi-dimensional (multi-indexed) - 2D photographic images,
hyperspectral images, etc.
financial, geographical, network (social networks), etc.
scalar signals, vector signals, etc.
Sk. Mohammadul Haque
Introduction & Review
Discrete-time Signals
Classification of Signals
Dimensionality:
One-dimensional: one independent variable. (e.g. f (t) defines
ambient temperature)
Multi-dimensional: two or more independent variables. (e.g.
f (x , y ) defines brightness of sky (How?))
Type of the domain/indices:
Continuous: Independent variables are continuous and can be
mapped to an interval in the real line (R).
Discrete: Independent variables are discrete and can be
mapped to a subset of the integers (Z).
Range:
Analog: Continuous-valued (real/complex) signals
Digital: Discrete-value signals.
Sk. Mohammadul Haque
Introduction & Review
Signals
Sk. Mohammadul Haque
Introduction & Review
Signals-Energy
Consider a discrete-time signal x [n] defined over discrete time
index n.
Energy of the signal, Ex (n1 , n2 ) between times n1 and n2 is
defined as
Ex (n1 , n2 ) =
n2
X
|x [n]|2 .
n=n1
Sk. Mohammadul Haque
Introduction & Review
Signals- Average Power
Consider a discrete-time signal x [n] defined over discrete time
index n.
Average Power of the signal, Px (n1 , n2 ) between times n1 and
n2 is defined as
Px (n1 , n2 ) =
Sk. Mohammadul Haque
Ex (n1 , n2 )
.
(n2 − n1 + 1)
Introduction & Review
Signals- Average Power
Energy signals - Signals having finite energy.
Power signals - Signals having finite average power.
Other than above - Signals that are neither energy signals nor
power signals.
Sk. Mohammadul Haque
Introduction & Review
Signals
Symmetry
Even signals: x [−n] = x [n].
Odd signals: x [−n] = −x [n].
A simple signal analysis
Given a signal y [n], we can decompose it as a sum of an even
signal e[n] and an odd signal o[n]
y [n] = e[n] + o[n]
1
1
e[n] = (y [n] + y [−n]) and o[n] = (y [n] − y [−n]).
2
2
can do much more interpretation.
Sk. Mohammadul Haque
Introduction & Review
Signals
Conjugate symmetry
Conjugate Symmetry: x [−n] = x ∗ [n].
Conjugate Anti-symmetry: x [−n] = −x ∗ [n].
Examples? e.g. for complex-valued signals?
Sk. Mohammadul Haque
Introduction & Review
Signals
Symmetry & Conjugate symmetry
Are they closed under addition operation, scalar
multiplications?
Examples?
Direct-sum of odd signals and even signals? Let’s discuss very
briefly.
Sk. Mohammadul Haque
Introduction & Review
Signals
Some useful signals
exponential signals
sinusoidal signals
step signals
impulse signals
rectangle signals
ramp signals
Sk. Mohammadul Haque
Introduction & Review
Bounded Signals
Bounded signals are of the form:
Discrete-time: x [n], if an M such that |x [n]| ≤ M for all n.
e.g. x [n] = sin 10n − 5.
How useful?
Sk. Mohammadul Haque
Introduction & Review
Periodic Signals
Periodic signals are of the form:
Discrete-time: x [n], if there exists a positive integer N such
that x [n] = x [n + N] .
x [n] has a period N .
e.g. x [n] = n − 11 · floor
n
.
11
What about x [n] = 2 sin (10n + 5)?
Sk. Mohammadul Haque
Introduction & Review
Periodic Signals
Fundamental period - Smallest period of a periodic signal.
If the period of a periodic signal is T , then kT for all positive
integer k is also a period.
Signals that not periodic are called aperiodic signals.
Sk. Mohammadul Haque
Introduction & Review
Periodic Signals
Are these periodic?
Sk. Mohammadul Haque
Introduction & Review
Exponential Signals
Exponential signals are of the form:
Continuous-time : x (t) = Ae bt = Ac t (very general)
If both A and b are real, then we have continuous-time real
exponential.
Discrete-time: x [n] = Ae bn = Ac n (very general)
If both A and b are real, then we have discrete-time real
exponential.
How useful? Bounded? Periodic?
Sk. Mohammadul Haque
Introduction & Review
Sinusoidal Signals
Real sinusoidal signals are of the form :
Discrete-time: x [n] = A cos (Ωn + δ) and A, Ω, δ are reals.
x1 [n] = 10 cos (5n + 4).
How useful? Bounded? Periodic?
Sk. Mohammadul Haque
Introduction & Review
Sinusoidal Signals
Real sinusoidal signals:
Sk. Mohammadul Haque
Introduction & Review
Sinusoidal Signals
Complex sinusoidal signals are of the form :
Discrete-time: x [n] = Ae bn = Ac n and b = jΩ is imaginary for
some real Ω.
jΩn
jΩn
jδ
jδ
jΩn
x [n] = Ae
= |A| e e
= |A| e · e
|A| (cos (Ωn + δ) + j sin (Ωn + δ)).
How useful? Have we come across the above?
Bounded? Periodic?
Sk. Mohammadul Haque
Introduction & Review
=
Unit Step Signals
Discrete-time unit step signal:
u[n] =
1,
n≥0
0, otherwise
Sk. Mohammadul Haque
Introduction & Review
Unit Step Signals
.
Sk. Mohammadul Haque
Introduction & Review
Unit Impulse Signals
Discrete-time unit impulse signal:
δ[n] =
1,
n=0
0, otherwise
Sk. Mohammadul Haque
.
Introduction & Review
Unit Impulse Signals
.
Sk. Mohammadul Haque
Introduction & Review
Transformations of Signals
Time-shifting
Given a discrete-time signal x [n], we have y [n] = x [n − b] for a
given integer b.
When will the signals shift towards left? Or, towards right?
Delayed or advanced?
Sk. Mohammadul Haque
Introduction & Review
Transformations of Signals
Time-shifting
Sk. Mohammadul Haque
Introduction & Review
Transformations of Signals
Time-reversal
Given a discrete-time signal x [n], we have y [n] = x [−n].
Sk. Mohammadul Haque
Introduction & Review
Transformations of Signals
Time-reversal
Sk. Mohammadul Haque
Introduction & Review
Transformations of Signals
Time-scaling
Given a discrete-time signal x [n], does this make sense:
y [n] = x [bn] for a given real positive b (?).
We will see later downsampling and upsampling in
discrete-time domain.
Sk. Mohammadul Haque
Introduction & Review
Unit Impulse Signals - Properties
Discrete-time unit impulse signal:
δ[n] =
1,
n=0
0, otherwise
.
Similar to continuous-time domain, we have
∞
X
n=−∞
∞
X
f [n]δ[n] = f [0]
f [n]δ[n − n0 ] = f [n0 ] for all integer n0 (sifting/sampling
n=−∞
property)
f [n]δ[n − n0 ] = f [n0 ]δ[n − n0 ] (equivalence property).
Sk. Mohammadul Haque
Introduction & Review
Discrete-time Systems
Discrete-time Systems
are mathematical entities (functions/process) of a number of
input signals and a number of output signals. We write, in
general, y [n] = H2 (x [n]).
behave as what are called ’functionals’ i.e. map functions
(signals) to functions (signals).
Simple examples: y [n] = 10x [n]
Here x [n] is the input signals and y [n] := B(x [n]) are the
output signals.
Sk. Mohammadul Haque
Introduction & Review
Systems
Classification of Systems
Continuous-time systems: Where both input and output
signals are continuous-time signals.
Discrete-time systems: Where both input and output signals
are discrete-time signals.
Mixed discrete-continuous systems.
Other systems with non-time variable signals are generalised
versions.
Sk. Mohammadul Haque
Introduction & Review
Systems
Discrete-time Systems - examples
Digital communication systems
signals - message
systems - transmitter, channel, receiver
Digital control systems
signals - reference inputs, sensor outputs, position, velocity,
temperature
systems - controller, plant
Digital signal conditioning systems
signals - audio, video
systems - filters
Sk. Mohammadul Haque
Introduction & Review
Systems - Shift Invariance
Shift (Time) Invariance:
Shift-Invariant, if for every input causing some output, every
shifted version by an amount a of the input produces the
same output but shifted by the amount a.
Discrete-time systems:
i.e. for system H1 , any input x [n] causes the output y [n],
we have for the shifted input x [n − a] causing the output
y [n − a].
y[n] = H1 (x[n])⇒y[n − a] = H1 (x[n − a]) for all x , n, a.
Otherwise, we call the system shift-variant (time-variant).
Sk. Mohammadul Haque
Introduction & Review
Systems - Linearity
Linear Systems: A system H is linear if for all input signals
x 1 and x 2 and scalar α, the following hold true:
H(x 1 + x 2 ) = H(x 1 ) + H(x 2 ). (Additivity property)
H(αx 1 ) = αH(x 1 ). (Homogeneity property)
Linearity is same as super-position property.
Example: y [n] = 2x [n].
Sk. Mohammadul Haque
Introduction & Review
Systems - Response
To learn about a given system,
we may want to investigate how it behaves for a chosen set
of input signals.
Which chosen set?
Let’s try with unit impulse signal - as it is extremely
localised?
Sk. Mohammadul Haque
Introduction & Review
Systems - Response
Recall the sifting property:
∞
X
f [n] =
f [m]δ[m − n].
m=−∞
[Obvious] If we have the samples f [m], we can synthesise the
original signal f [n], by taking linear combinations of the
shifted impulses.
But will the shifted impulses make the system behave
similarly individually?
What about Shift-Invariance?
Sk. Mohammadul Haque
Introduction & Review
LSI (LTI) Systems
Recall the sifting property:
f [n] =
∞
X
f [m]δ[n − m].
m=−∞
Input signal = Superposition of scaled impulses.
Linear Shift (Time)-Invariant Systems:
Systems which obey the linearity property and the shift
(time)-invariance property.
Sk. Mohammadul Haque
Introduction & Review
LSI (LTI) Systems - Response
Discrete-time domain:
For any input x [n] and system H, we have the output y [n] as
y [n]
=
=
H(x [n])
∞
X
H(
x [m]δ[n − m])
m=−∞
=
=
=
∞
X
m=−∞
∞
X
m=−∞
∞
X
H(x [m]δ[n − m]),
[additivity]
x [m] · [H(δ[n − m])],
[homogeneity]
x [m] · [hm [n]]
m=−∞
···
=
·····················
∞
X
x [m] · h[n − m],
[shift-invariance].
m=−∞
Sk. Mohammadul Haque
Introduction & Review
Convolution
Discrete-time domain:
For any input x [n] and system H, we have the output y [n] as
y [n]
=
∞
X
x [m] · h[n − m].
m=−∞
If x [n] = δ[n], we have y [n] = h[n]. Hence, the sequence h[n]
is called its unit impulse response.
Sk. Mohammadul Haque
Introduction & Review
Convolution
Convolution Sum:
y [n] =
∞
X
x [m] · h[n − m] =: x [n] ∗ h[n] =: (x ∗ h)[n].
m=−∞
Sk. Mohammadul Haque
Introduction & Review
Convolution - Properties
Associative:
(x [n] ∗ h1 [n]) ∗ h2 [n] = x [n] ∗ (h1 [n] ∗ h2 [n]) .
Sk. Mohammadul Haque
Introduction & Review
Convolution - Properties
Commutative:
h1 [n] ∗ h2 [n] = h2 [n] ∗ h1 [n].
Sk. Mohammadul Haque
Introduction & Review
Convolution - Properties
Distributive:
x [n] ∗ (h1 [n] + h2 [n]) = x [n] ∗ h1 [n] + x [n] ∗ h2 [n].
Sk. Mohammadul Haque
Introduction & Review
LSI (LTI) Systems - Causality
Convolution:
y [n] =
∞
X
x [m] · h[n − m] =: x [n] ∗ h[n].
m=−∞
Causality:
For a system with impulse response h, we have
h[n]
Sk. Mohammadul Haque
= 0, when n < 0.
Introduction & Review
LSI (LTI) Systems - Stability I
For bounded input, output is also bounded.
Discrete-time domain: (Input bound is B x )
y [n] =
|y [n]| =
≤
∞
X
x [m] · h[n − m]du =: x [n] ∗ h[n].
m=−∞
∞
X
x [m] · h[n − m]
m=−∞
∞
X
|x [m] · h[n − m]| =
m=−∞
≤ Bx ·
∞
X
∞
X
|x [m]| · |h[n − m]|
m=−∞
∞
X
|h[n − m]| = Bx ·
m=−∞
|h[m]| .
m=−∞
So, if impulse
P∞response h[m] is absolutely summable
(∥h∥1 := m=−∞ |h[m])| < ∞), we have BIBO stability.
Sk. Mohammadul Haque
Introduction & Review
LSI (LTI) Systems - Stability II
For bounded input, output is also bounded.
h(u) is absolutely integrable and h[m] is absolutely
summable is sufficient to show BIBO stability.
Are they necessary?
e.g. If we assume
∞
X
|h[m]| = ∞ is not finite, can we come
m=−∞
up with a bounded input x [n] such that
(x [m] · h[m − n]) = |h[m − n]| for all m, but, at least one
value of n?
Sk. Mohammadul Haque
Introduction & Review
LSI (LTI) Systems - Memory
Convolution:
y [n]
=
∞
X
x [m] · h[n − m] =: x [n] ∗ h[n].
m=−∞
Memoryless:
For an LTI system with impulse response h, we have
a)
h[n]
=
0, when n ̸= 0.
b)
h[0]
=
K′
and for some constants K and K ′ .
Sk. Mohammadul Haque
Introduction & Review
LSI (LTI) Systems - Invertibility
Forward System H1
Represented by convolution.
How to find invertibility?
How about H1 ∗ H2 = Id ?
h1 [n] ∗ h2 [n] = δ[n]
Sk. Mohammadul Haque
Introduction & Review
Linear Constant Co-efficient Difference Equations
In general, we have relationship:
N
X
M
X
ak y [n − k] =
k=0
bk x [n − k]
k=0
Initial conditions: y [n0 ], y [n0 − 1], · · · , y [n0 − N + 1].
Order is N. Solution has two parts:
One due to homogeneous
N
X
ak y [n − k] = 0. (Natural
k=0
response)
Other explicitly due to input x [n]. (Forced response /
Particular solution)
Conditions for LTI? Causal?
Sk. Mohammadul Haque
Introduction & Review
Motivation
Desirable properties for synthesis and analysis:
Set of basic signals (done earlier in a way!)
Not just input signals, but also output signals to be simple.
Recall LSI (LTI) system process:
y [n]
= x [n] ∗ h[n]
∞
X
x [m] · h[n − m].
m=−∞
Sk. Mohammadul Haque
Introduction & Review
Motivation
Discrete-time:
y [n]
= x [n] ∗ h[n] =
∞
X
x [m] · h[n − m].
m=−∞
Set input x [n] = z n . We have output as
y [n]
∞
X
=
=
=
m=−∞
∞
X
z m · h[n − m]
z n−m · h[m]
m=−∞
∞
X
n
z −m · h[m]
z
m=−∞
{z
|
=
H(z)
}
H(z)z n .
Sk. Mohammadul Haque
Introduction & Review
Motivation
Discrete-time:
We have output as y [z] = H(z)z n .
Output signal is scalar multiple of input signal.
Scalar H(z) is a function of the parameter z.
Signals z n are the eigensignals
(eigenfunctions/eigensequences) for the LSI (LTI) system.
The scalars H(z) are called the eigenvalues corresponding to
the eigensignals z n .
Sk. Mohammadul Haque
Introduction & Review
Induced Gains in DT Systems
Now, we can represent the DT LSI process as y = Sx.
(However, S linear system matrix can be, in general, any
square N × N matrix. Further even, if we still generalise to
non-linear systems, we can simply write y = S (x) where
S : CN → CN any general function.)
We define the “vector p-norm” ∥v∥p , 1 ≤ p ≤ ∞ for any
N-length complex vector v as follows:

1

!


N−1

X
p



,
for 1 ≤ p < ∞
|vi |p


i=0
∥v∥p :=
1


!

N−1

X


p p


lim
|vi |
,
p→∞
for p = ∞.
i=0
Sk. Mohammadul Haque
Introduction & Review
Induced Gains in DT Systems
We define the “induced matrix (system) p-norm”
|∥S∥|p , 1 ≤ p ≤ ∞ for any LSI matrix (system) S as follows:
|∥S∥|p :=
sup
x,∥x∥p =1
∥S (x)∥p = sup
∥S (x)∥p
x̸=0
∥x∥p
.
The above definition finds the induced gain of the system S.
That is, for all input signals whose p-norms are bounded, we
seek what is the maximum (or supremum) p-norm among
the corresponding outputs. If |∥S∥|p < ∞, we say the system
is induced p-norm (gain) stable. In particular, if the product
|∥S∥|p · S−1 p is near unity, we say that the system S is
well-conditioned, otherwise, we say that the system S is
ill-conditioned.
Sk. Mohammadul Haque
Introduction & Review
Discrete-time - Motivation
Discrete-time Fourier Series:
Forward Transform: ck =
N−1
1 X
N
Inverse Transform: h[n] =
2π
−jk
h[n]e
n
N .
n=0
N−1
X
2π
jk
ck e
n
N .
k=0
We may want to consider aperiodic signals where N → ∞
or with time-shifting N1 , N2 → ∞ if the period runs from −N1
to N2 and N1 + N2 + 1 = N. As also, we want to ensure that
2πk
= ωk remains finite, so also, sweeps continuously
N
over all real frequencies ω ∈ R.
Then, we have
δωk
ωk+1 − ωk
(k + 1) − (k)
1
=
=
= .
2π
2π
N
N
Sk. Mohammadul Haque
Introduction & Review
Discrete-time - Motivation
Putting ωk and
δωk
, we get
2π
ck
N2
δωk X
=
2π
N−1
X
h[n] =
h[n]e −j ωk n
n=−N1
ck e j ωk n .
k=0
We now only need a rescaling in definition of ck to cˇk as
below:
cˇk
=
N2
X
h[n]e −j ωk n
n=−N1
h[n] =
Sk. Mohammadul Haque
X
δωk N−1
2π
cˇk e j ωk n .
k=0
Introduction & Review
Discrete-time Fourier Transform
We define ω, dω, H(e jω ) := lim (ωk , δωk , cˇk ). Then, in
N→∞
the limit as N, N1 , N2 → ∞, we have
jω
H(e ) =
h[n] =
∞
X
h[n]e −j ωn
n=−∞
ˆ
1 2π
2π
H(e j ω )e j ωn · d ω.
0
The above H(e j ω ) is called the Fourier Transform of h[n].
H(e j ω ) is also denoted as Fn (h[n]) where the transform is
perform along the n dimension. If clear from context, we
simply write F(h[n]). Similarly, for the inverse transform
jω
−1 (H(e j ω )).
F−1
ω (H(e )) as F
Sk. Mohammadul Haque
Introduction & Review
Discrete-time Fourier Transform
H(e j ω ) =
∞
X
h[n]e −j ωn is called the forward Fourier
n=−∞
transform (analysis) equation.
ˆ
1 2π
H(e j ω )e j ωn · d ω is called the inverse Fourier
h[n] =
2π 0
transform (synthesis) equation.
DT F T
We denote the above as h[n] ←−−→ H(e j ω ).
Periodicity of DTFT
Note that
H(e
j (ω+2π)
)=
∞
X
h[n]e
n=−∞
−j (ω+2π)n
∞
X
=
1
h[n]e
−j ωn
n=−∞
or H(e j (ω+2π) ) = H(e j ω ). Hence, DTFT H(e j ω ) is
2π-periodic. (Differs from CTFT).
Sk. Mohammadul Haque
Introduction & Review
z }| {
e −j 2πn
Convergence - DTFT
Convergence to equality
Finite energy signals (Sufficient condition- MSE sense)
A signal h[n] with finite energy E =
X∞
n=−∞
2
|h[n]| < ∞.
or
Absolutely summable signals (Sufficient conditionuniform convergence)
A signal h[n] with absolute sum S =
Sk. Mohammadul Haque
X∞
n=−∞
Introduction & Review
|h[n]| < ∞.
Discrete-time Fourier Transform
DTFT – Properties:
DT F T
DT F T
Linearity: h1 [n] ←−−→ H1 (e jω ) and h2 [n] ←−−→ H2 (e jω )
DT F T
implies αh1 [n] + βh2 [n] ←−−→ αH1 (e jω ) + βH2 (e jω ).
DT F T
Shifting: h[n] ←−−→ H(e jω ) implies
DT F T
h[n − n0 ] ←−−→ e −jωn0 H(e jω ) and
DT F T
e jω0 n h[n] ←−−→ H(e j(ω−ω0 ) ).
DT F T
Time-reversal: h[n] ←−−→ H(e jω ) implies
DT F T
h[−n] ←−−→ H(e −jω ).
DT F T
Time-scaling: h[n] ←−−→ H(e jω ) implies
DT F T
h[an] ←−−→ H(e jaω ) (zero-filled scaling).
Sk. Mohammadul Haque
Introduction & Review
Discrete-time Fourier Transform
DTFT – Properties:
DT F T
Multiplication: h1 [n] ←−−→ H1 (e jω ) and
DT F T
h2 [n] ←−−→ H2 (e jω ) implies
1
DT F T
h1 [n]h2 [n] ←−−→
H1 (e jω ) ∗ H2 (e jω ).
2π
DT F T
Convolution: h1 [n] ←−−→ H1 (e jω ) and
DT F T
h2 [n] ←−−→ H2 (e jω ) implies
DT F T
h1 [n] ∗ h2 [n] ←−−→ H1 (e jω )H2 (e jω ).
DT F T
Complex-conjugation: h[n] ←−−→ H(e jω ) implies
DT F T
h∗ [n] ←−−→ H ∗ (e −jω ).
DT F T
Real signal: h[n] ←−−→ H(e jω ) implies H(e jω ) = H ∗ (e −jω ).
DT F T
Real and even signal: h[n] ←−−→ H(e jω ) implies H(e jω )
real and even.
DT F T
Real and odd signal: h[n] ←−−→ H(e jω ) implies H(e jω )
imaginary and odd.
Sk. Mohammadul Haque
Introduction & Review
Discrete-time Fourier Transform
DTFT – Properties:
Parseval’s Relation: ˆ
X∞
1
2
|h[n]| =
n=−∞
2π
2π
2
H(e jω ) · dω.
0
DT F T
Difference: h[n] ←−−→ H(e jω ) implies
DT F T
h[n] − h[n − 1] ←−−→ (1 − e −jω )H(e jω ).
DT F T
Summation: h[n] ←−−→ H(e jω ) implies
Pn
DT F T
m=−∞ h[m] ←−−→
P∞
1
H(e jω ) + πH(0) k=−∞ δ(ω − 2πk).
(1 − e −jω )
Sk. Mohammadul Haque
Introduction & Review
Discrete-time Fourier Transform
DTFT – Periodic Signals:
Consider a periodic signal h[n] with period N.
Recall the DT Fourier Series of h[n] =
N−1
X
ck e jnω1 k .
k=0
Find Fourier transform of each component ck e jnω1 k as
P
DT F T
ck e jnω1 k ←−−→ 2πck ∞
m=−∞ δ(ω − kω1 − 2πm).
Hence, we have
N−1
X
DT F T
ck e jnω1 k ←−−→
PN−1
k=0
2πck
P∞
m=−∞ δ(ω
k=0
Sk. Mohammadul Haque
Introduction & Review
− kω1 − 2πm) .
Discrete-time Fourier Transform
DTFT – Periodic Signals:
Consider a periodic signal h[n] with period N.
Recall the DT Fourier Series of h[n] =
N−1
X
ck e jnω1 k .
k=0
Find Fourier transform of each component ck e jnω1 k as
P
DT F T
ck e jnω1 k ←−−→ 2πck ∞
m=−∞ δ(ω − kω1 − 2πm).
Hence, we have
N−1
X
DT F T
ck e jnω1 k ←−−→
PN−1
k=0
2πck
P∞
m=−∞ δ(ω
k=0
Sk. Mohammadul Haque
Introduction & Review
− kω1 − 2πm) .