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DSP Lecture
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Digital Signal Processing
Spring 2012
Lecture 2: Discrete-Time Signals & Systems
Instructor:
Engr. Asif Iqbal
Department of Electrical Engineering
NUCES-FAST, Peshawar
asif.iqbal@nu.edu.pk
Part 1: Discrete-Time Signals
• Sequence of numbers
, ∞ ∞
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where n is an integer
• Periodic sampling of an analog
Digital Signal Processing, Lecture 2,
Spring 2012
signal
, ∞ ∞
where T is the sampling period
2
Sequence Operations
• The product and sum of two sequences x[n] and y[n]: sample-
by-sample product and sum, respectively.
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• Multiplication of a sequence x[n] by a number :
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multiplication of each sample value by .
• Delay or shift of a sequence x[n]
,
where is an integer
3
Basic Sequences
• Unit Sample Sequence (discrete-time Impulse or Impulse)
0, 0, Unit Sample
1, 0.
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... ...
Digital Signal Processing, Lecture 2,
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• Any sequence can be represented as a sum of scaled, delayed
impulses
3 1 3,
• More generally,
∑!"
4
Unit Step Sequence
• Defined as Unit Step
1, $ 0,
#
0, 0.
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... ...
Digital Signal Processing, Lecture 2,
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• Related to the Impluse by
# ∑%
!" ,
# 1 2 ∙∙∙ or
# ∑!"
• Conversely
# # 1 5
Exponential Sequences
• These are very important in representing and analyzing LTI
discrete-time systems. Real
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• Defined as Exponential
( % ... ...
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• If A and are real numbers, the sequence is real.
• If 0 1 and A is positive, the sequence values are
positive and decrease with increasing n.
• If 0 ) ) 1 , the sequence values alternate in sign, but
again decrease in magnitude with increasing n.
• If ) 1 , the sequence values increase with increasing n.
6
Combining Basic Sequences
• An exponential sequence that is zero for 0
( % , ≥ 0
=
0, < 0
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Digital Signal Processing, Lecture 2,
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• A much simpler expression
= ( % #[ ]
7
Sinusoidal Sequences
• A general sinusoidal function
( cos - . for all
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With ( and . are real constants
• The ( % with complex has real and imaginary parts that are
Digital Signal Processing, Lecture 2,
Spring 2012
exponentially weighted sinusoids.
If / 012 and ( ( / 03 , then
%
=( = ( / 03 / 012 % ,
= ( /0 12 %43
,
= ( cos - + . + j sin - +.
8
Complex Exponential Sequence
• When 1,
= ( /0 12 %43
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= ( cos - + . + j sin - +.
• By analogy with the continuous-time case, the quantity - is
Digital Signal Processing, Lecture 2,
Spring 2012
called the frequency of the complex sinusoid or complex
exponential and . is called the phase.
• is always an integer (differences between discrete-time and
continuous-time)
9
An important difference:
Frequency Range
• Consider a frequency - 28
(/ 0 12 49: % (/ 012 % / 09:% (/ 012 % ,
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• More generally, - + 28; , ; being an integer
= (/ 0 12 49:< % = (/ 012 % / 09:<% = (/ 012 % ,
Digital Signal Processing, Lecture 2,
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• Same for sinusoidal sequences
= ( cos - + 28; (
+ . = ( cos - + .)
• So, only consider frequencies in an interval of 28, such as
−8 < - ≤ 8 or 0 ≤ - < 28
10
Another important difference:
Periodicity
• In the continuous-time case, a sinusoidal signal and a complex
exponential signal are both periodic.
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• In the discrete-time case, a periodic sequence is defined as
>, for all
Digital Signal Processing, Lecture 2,
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where the period N is necessarily an integer.
• For sinusoid,
( cos - (
+ . = ( cos - + - > + .)
9:!
which requires that - > = 28 ?; > =
12
where is an integer 11
Another important difference:
Periodicity
• Same for complex exponential sequence
/ 012 %4@ / 012 %
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which is true for - 28
• So, complex exponential and sinusoidal sequences
Digital Signal Processing, Lecture 2,
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• are not necessarily periodic in with period 28/-
• and, depending on the value of - , may not be periodic at all.
• Consider
:%
• cos , with a period of N=8
B
:%
• 9 = cos , with a period of N=16
C
• Increase in frequency => Increase in period!
12
Another important difference:
Frequency
• For a continuous-time sinusoidal signal
D ( cos Ω D . ,
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as Ω increases, D oscillates more and more rapidly
Digital Signal Processing, Lecture 2,
Spring 2012
• For the discrete-time sinusoidal signal
( cos - .
• As - increases from 0 towards 8, oscillates more and more
rapidly
• As - increases from 8 towards 28, the oscillations become
slower
13
Part 2: Discrete-Time Systems
• A transformation or operator that maps input into output
F G
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Digital Signal Processing, Lecture 2,
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• Examples:
• The ideal delay system
H , ∞ ∞
• A memory less system
9,
∞ ∞
14
Linear Systems
Digital Signal Processing, Lecture 2,
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15
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• a
Time-invariant Systems
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16
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Causality
• The output sequence value at the index depends only
on the input sequence values for ≤
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• Example
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H , −∞ < <∞
• Causal for H >0
• Non-causal for H <0
17
Stability
• A system is stable in the BIBO sense if and only if every
bounded input sequence produces a bounded output
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sequence.
Digital Signal Processing, Lecture 2,
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• Example
9,
−∞ < <∞
18
