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Om Sakthi
Adhiparasakthi Engineering CollegeMelmaruvathur – 603319
CS2403 Digital Signal Processing
Lecture Notes by
R. Sivarajan, AP/ECE/APEC
Department of Electronics and Communication
Engineering
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Unit – I Signals and Systems
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Basic Elements of DSP
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Contd…
Advantages of DSP over ASP:Stable, reliable, flexible,
predictable, repeatable
Choose any accuracy by increasing or decreasing number of
bits
Sharing of digital processor
Achieve linear phase characteristics
Multi rate processing is possible
Digital circuits connected in cascade without any
loadingproblem
Storage of digital data very easy
For processing low frequency signal (seismic signal),
analogcircuits requires inductor and capacitor of very large size,
so
we prefer digital processor for such application
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Contd…
Disadvantages of DSP over ASP:Needs pre and post processing (ADC
& DAC)
Suffer from frequency limitation
Analog circuits don’t need much power where digital circuit
needs more power consumption
Applications of DSP:
Telecommunication
MilitaryConsumer Electronics
Instrumentation and Control
SeismologyImage processing
Speech processing
Medicine, Signal filtering
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Concept of Frequency in Analog and Digital Signal
Continuous time sinusoid signal Simple harmonic oscillation
defined by sinusoid signal as = + ,−∞ < <
∞ A – amplitude of sinusoids; Ω – frequency in rad/sec; θ
–
phase in radians and related as =
2 Thus,
= 2 + ,−∞ < < ∞
Properties of CT sinusoid signal is periodic;
= + , = 1 ⁄ , is afundamental period of
sinusoids
CT sinusoid signal with distinct frequency themselves
differIncrease in F result in increase in rate of oscillation of
the
signal
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Contd…
Relationship of sinusoid signal in terms of exponential
signal
= 2 + 2 ,−∞ < <
∞ Sinusoidal signal adding two equal amplitude
complex
conjugate exponential signal
Positive frequency counter clockwise uniform angular
motion
Negative frequency clockwise uniform angular motion
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Contd…
Discrete time sinusoid signal The DT sinusoid signal
expressed as = + , −∞ < < ∞A – amplitude of sinusoids;
ω – frequency in rad/samples; θ
– phase in radians; and related as =
2 Thus, = 2 + , −∞ < < ∞ Properties of
DT sinusoid signal
is periodic only if its frequency is rational
number; + = , smallest value of is fundamental periodDT
sinusoids whose frequency separated by an integer
multiples of 2π are identicalhe highest rate of oscillation in a
discrete time sinusoids is
attained when = (or = −) or equivalently =
(or = −
)
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Contd…
Relationship of sinusoid signal in terms of exponential
signal
= 2 + 2 ,−∞ < < ∞Sinusoidal
signal adding two equal amplitude complex
conjugate exponential signal
Positive frequency counter clockwise uniform angular
motion
Negative frequency clockwise uniform angular motion
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Sampling Theorem
Sampling continuous time to discrete time signal
Sampling performed by taking samples of CT signal at
definite interval of time
Time interval between successive samples sampling
time
Inverse of sampling period sampling frequency Fs. =
/ If highest frequency of analog signal is is
= andsignal is sampled at > 2 ≈ 2, then
can easilyextracted from its sample value using
interpolation function
= 22 Sampling rate
= 2 = 2, Nyquist rate
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Discrete Time Signals
Representation of Signals
Functional representation
Tabular representation
Sequence representation
Graphical representation
Some elementary DT Signals:
Unit sample sequence = 1 = 0; = 0 ≠ 0Unit step sequence
= 1 ≥ 0; = 0 < 0Unit ramp sequence = ≥ 0; = 0 <
0
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Contd…
Exponential sequence: =
> 1 Grows exponentially
< < 1
Decays exponentially < −1 Grows exponentially;
Alternates between +ve and –ve
− < < 0 Decays exponentially; Alternates between +ve
and –ve
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Contd…
Discrete Time Sinusoid Signal:
= + ; −∞ < < +∞
= + ; −∞ < < +∞
– frequency in radians/ sampleθ – phase in radians
= 2
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Contd…
Deterministic Signal: Nature and amplitude of signal
can
be predicted
Random Signal: Nature and amplitude of signal cannot be
predicted
Periodic Signal: + = , −∞ < < ∞ Aperiodic Signal: + ≠
, −∞ < < ∞ Even Signal:
= −
Odd Signal: = −− Even component of signal: = +
− Odd component of signal:
= − −
Causal Signal: Right Sided Sequence, = 0, < 0, defined
at ≥ 0;Non Causal Signal: Two Sided Sequence,
defined at
both ≥ 0 and > 0;
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Contd…
Anti Causal Signal: Left Sided Sequence, defined at ≤
0. Energy and Power Signal:
Energy signal: finite energy and zero average power
Power signal: infinite energy and finite average
power
Energy of the signal:
= lim→ || Power of the
Signal:
= lim→ 12 + 1 ||
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Discrete Time Systems
Representation of Systems:
Block diagram representationSignal flow graph
Element Block Diagram
Representation
Signal flow Graph
Adder
Constant Multiplier
Unit delay element
Unit advance
element
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Classification of Systems
Static System:Output depends on present input not on past or
future inputNo memoryDynamic System:
Output depends on both present and past inputHas a memoryLinear
System: Superposition principle holds
H[a*x(n)+b*y(n)]=a*H[x(n)]+b*H[y(n)]
Non Linear System: Superposition principle does not holdsTime
Inariant System: Input Output relationship does notvary with
time
H[x(n-k)]=y(n-k)
Time Variant System: Input Output relationship vary withtime
Linear Time Invariant (LTI) System: System which satisfy
both linearity and time invariant condition called LTI
System
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Contd…
Causal System: Output depends only on present and pastinput and
does not depends on future inputNon Causal System: Output depends
on present, past and
future input
Stable and Unstable System: System said to be bounded input
bounded output (BIBO)stable, if every bounded input produces
bounded outputBounded signal has amplitude which remains finite
BIBO stable system produces bounded output for anybounded input
so that it does not grow unreasonable largeConditions: If
system transfer function is a rational fraction, then degree
of numerator must no longer than degree of denominatorPoles lie
in left half plane of S – plane or within unit circle inZ – planeNo
repeated poles lie on the imaginary axis
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Analysis of Discrete Time LTI Systems
Techniques for the analysis of linear system:
Input output equation for system
= − 1, − 2, , − , , − 1, , − For an LTI system, input
output relationship can be
expressed as
= −
− −
This input output relationship is called difference
equation
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Contd…
Response of LTI systems to arbitrary inputs: The
Convolution Sum:
= ℎ −
Properties of Convolution:
Commutative Property:
∗ ℎ = ℎ ∗ Associative Property:
[ ∗ ℎ] ∗ = ∗ [ℎ ∗ ]
Distributive Property: ∗ [ℎ + ] = [ ∗ ℎ] + [ ∗ ]
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Contd...
Condition for Stability of LTI Systems:
|ℎ|
< ∞
FIR System: Finite number of samples, Requires a memory
of length N, Described by a difference equation,
= − IIR System: Infinite number of
samples, Requires a infinitememory, Described by a difference
equation,
= − − −
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Z Transform
Two Sided Z Transform:
= [] =
∞
∞
One Sided Z Transform:
= [] = ∞
Inverse Z transform: = 12
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ROC
Definition:
ROC (Region of Convergence) of is the set of all
values of, for which attains a finite
value.Properties:
ROC of is a ring or disk in Z plane, with center at
origin.If is finite duration right sided (causal) signal,
then the ROCis entire Z plane except = 0.If
is finite duration left sided (anti causal) signal, then
the
ROC is entire Z plane except = ∞.If is finite duration
two sided (non causal) signal, then theROC is entire Z plane
except
= 0and
=∞.
If is infinite duration right sided (causal) signal, then
ROC isexterior of the circle of radius .If is infinite
duration left sided (anti causal) signal, then ROCis interior of
the circle of radius
.
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Contd…
If is infinite duration both sided (non causal) signal,
thenROC is the region in between two circles of radius and .If
is rational, then ROC does not include any poles of
.If is rational, and if is right sided
(causal), then ROC isexterior of the circle whose radius
corresponds to the pole withlargest magnitude.
If
is rational, and if
is left sided (anti causal), then ROC
is interior of the circle whose radius corresponds to the pole
withsmallest magnitude.
If is rational, and if is two sided (non
causal), then ROCis region in between two circles whose radius
corresponds to the
pole of causal part with largest magnitude and pole of anti
causal
with smallest magnitude.
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Properties of Z Transform
Linearity Property:
+ = [] + [] = + Proof:
[ + ] = [ + ]∞∞
[ + ] = ∞
∞ + ∞
∞
[ + ] =
∞
∞ +
∞
∞
[ + ] = +
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Contd…
Multiplication by exponential sequence
:
[] =
Proof:
[] = ∞
∞
[] = ∞
∞
[] = ∞
∞ [] =
[
] =
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Contd…
Multiplication by n:
[] = − Proof:
= ∞
∞
= ∞
∞
= ∞
∞ = −
∞
∞
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Contd…
− = ∞
∞
−
= []
[] = −
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Contd…
Shifting property:
If [] = , then[ − ] = [ + ] =
Proof:
[ − ] = −
∞
∞
Let = − , then = + If = −∞, then = −∞ If
= ∞, then
= ∞
[ − ] = ∞∞
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Contd…
[ − ] = ∞
∞
[ − ] = ∞
∞
[ − ] = Similarly, we can prove
[ − ] =
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Contd…
Convolution Theorem:
If [] = and [] = , then [ ∗
] = where,
∗ = −
Proof:
[ ∗ ] = [ ∗ ]∞
∞
[ ∗ ] = − ∞
∞
Let
= − , then
= +
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Contd…
[ ∗ ] = ∞
∞
[ ∗ ] =
∞
∞ [ ∗ ] =
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Contd…
Initial Value Theorem:
If [] = , then
0 = lim→
Proof:
= ∞
= 0 + 1
+ 2
+ ∞
Taking limit → ∞, lim→ = 0
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Contd…
Final Value Theorem:
If [] = , then∞ = lim→ 1 −
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Inverse Z Transform
Residue Method
Long Division MethodPartial Fraction Method
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Convolution
Linear Convolution
yn = xkhn − k
Circular Convolution:
= ○ = −
= ○ = − Various Methods of
calculating both Linear and Circular
Convolution between two sequences areGraphical Method
Tabular MethodMatrix method
C l i
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Correlation
Auto Correlation:
= − ∞
∞
Cross Correlation:
= − ∞
∞
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Unit – II Frequency Transformation
Di t F i t f
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Discrete Fourier transform
N point DFT
= ⁄
, = 0, 1, 2, , − 1
N point IDFT
= 1
⁄
, = 0, 1, 2, , − 1
Properties of DFT
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Properties of DFT
Periodicity property:
If is an N point DFT of , then + =
+ =
Linearity property:If and are the N point
DFT of and respectively, and and are
arbitrary constants either realor complex valued, then [ +
] = + Convolution property:
If
and
are the N point DFT of
and
respectively, then [ ∗ ] =
Contd
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Contd…
Time reversal property:
If is the N point DFT of the sequence ,
then
[ − ] = −
Circular time shift property:If is the DFT of the
sequence , then [ − , ] = ⁄ Circular frequency
shift property:If is the DFT of the sequence ,
then ⁄ = − , Circular convolution property of
DFT.If [] = and [] = , then[] = [⨂] =
Contd
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Contd…
Complex conjugate property:
If [] = , then[∗] = ∗−, = ∗ −
[ ∗] = 1 ∗ ⁄
= 1 ⁄
∗
Circular correlation property:
If
[] = and
[] = , then
[] = 1 = ∗ where,
= ∗ − ,
Contd
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Contd…
Multiplication of two sequences property:
If [] = and [] = , then [] =
[] = 1
[ ⨂]
Parseval’s property:For complex valued sequence and , if
[] = and [] = ,
∗ = 1 ∗ If = , then
|| = 1 | |
DITFFT Algorithm to calculate DFT
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DITFFT Algorithm to calculate DFT
DIFFFT Algorithm to Calculate DFT
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DIFFFT Algorithm to Calculate DFT
Filtering Method based on DFT
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Filtering Method based on DFT
– Input (Length M)ℎ – System impulse response (Length
N)
- Response of the system to an input (Length L =
M+N-1)
= ∗ ℎ Make length of and ℎ to M+N-1 by
appending zerosMake length of and ℎ in the order of
2 where is theinteger (length must be 2 = 4,
2 = 8, 2 = 16, etc)Find L point DFT of , i.e.,
Find L point DFT of
ℎ, i.e.,
Multiply and Find IDFT of [ ]
