# DSP Lecture by NumanSaeed

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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 :

Digital Signal Processing, Lecture 2,
Spring 2012
multiplication of each sample value by .

• Delay or shift of a sequence x[n]
,
where is an integer

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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,
Spring 2012
• Any sequence can be represented as a sum of scaled, delayed
impulses
3             1             3,
• More generally,
∑!"
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Unit Step Sequence
• Defined as                       Unit Step
1, 							 \$ 0,
#
0, 							   0.

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...                  ...

Digital Signal Processing, Lecture 2,
Spring 2012
• 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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Spring 2012
• 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.
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Combining Basic Sequences
• An exponential sequence that is zero for   0
( % , 								 ≥ 0
=
0, 														 < 0

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Digital Signal Processing, Lecture 2,
Spring 2012
• A much simpler expression
= ( % #[ ]

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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 -         +.
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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)

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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,
Spring 2012
• Same for sinusoidal sequences
= ( cos - + 28;                        (
+ . = (	cos	 -        + .)
• So, only consider frequencies in an interval of 28, such as
−8 < - ≤ 8 or         	0 ≤ - < 28

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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,
Spring 2012
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,
Spring 2012
• 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!
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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

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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,
Spring 2012
• Examples:
• The ideal delay system
H    , 																								 ∞    ∞
• A memory less system
9,
∞   ∞
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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

Digital Signal Processing, Lecture 2,
Spring 2012
H   , 																								 −∞ <   <∞

• Causal for   H   >0
• Non-causal for    H   <0

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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,
Spring 2012
• Example
9,
−∞ <   <∞

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