Transcript Document

Discrete-Time Signals and Systems
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there is no limit to its power in this field.
Paul Dirac
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Discrete-Time Signals: Sequences
• Discrete-time signals are represented by sequence of numbers
– The nth number in the sequence is represented with x[n]
• Often times sequences are obtained by sampling of
continuous-time signals
– In this case x[n] is value of the analog signal at xc(nT)
– Where T is the sampling period
10
0
-10
0
10
t (ms)
20
40
60
80
100
10
20
30
40
50 n (samples)
0
-10
0
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Basic Sequences and Operations
• Delaying (Shifting) a sequence
y[n]  x[n  no ]
• Unit sample (impulse) sequence
0 n  0
[n]  
1 n  0
• Unit step sequence
1.5
1
0.5
0
-10
0
5
10
-5
0
5
10
-5
0
5
10
1.5
0 n  0
u[n]  
1 n  0
• Exponential sequences
x[n]  An
1
0.5
0
-10
1
0.5
0
-10
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Sinusoidal Sequences
• Important class of sequences
xn  coson  
• An exponential sequence with complex    e jo and A  A e j
xn  A  A e  e
n
j
n
jon
 A  e jon  
n
xn  A  coson    j A  sinon  
n
n
• x[n] is a sum of weighted sinusoids
• Different from continuous-time, discrete-time sinusoids
– Have ambiguity of 2k in frequency
coso  2k n    coson  
– Are not necessary periodic with 2/o
coson    coson  oN   only if N 
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2k
is an integer
o
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Demo
Rotating Phasors Demo
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Discrete-Time Systems
• Discrete-Time Sequence is a mathematical operation that
maps a given input sequence x[n] into an output sequence
y[n]
y[n]  T{x[n]}
x[n]
T{.}
y[n]
• Example Discrete-Time Systems
– Moving (Running) Average
y[n]  x[n]  x[n  1]  x[n  2]  x[n  3]
– Maximum
y[n]  maxx[n], x[n  1], x[n  2]
– Ideal Delay System
y[n]  x[n  no ]
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Memoryless System
• Memoryless System
– A system is memoryless if the output y[n] at every value of n
depends only on the input x[n] at the same value of n
• Example Memoryless Systems
– Square
y[n]  x[n]
2
– Sign
y[n]  signx[n]
• Counter Example
– Ideal Delay System
y[n]  x[n  no ]
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Linear Systems
• Linear System: A system is linear if and only if
T{x1[n]  x2[n]}  Tx1[n]  Tx2[n] (additivity)
and
Tax[n]  aTx[n] (scaling)
• Examples
– Ideal Delay System
y[n]  x[n  no ]
T{x1[n]  x2[n]}
T{x2[n]}  Tx1[n]
Tax[n]
aTx[n]
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 x1[n  no ]  x2[n  no ]
 x1[n  no ]  x2[n  no ]

ax1[n  no ]

ax1[n  no ]
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Time-Invariant Systems
• Time-Invariant (shift-invariant) Systems
– A time shift at the input causes corresponding time-shift at output
y[n]  T{x[n]}  y[n  no ]  Tx[n  no ]
• Example
– Square
Delay the input the output is
y[n]  x[n]
2
Delay the output gives
y1 n  x[n  no ]
2
yn - no   x[n  no ]
2
• Counter Example
– Compressor System
y[n]  x[Mn]
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Delay the input the output is
Delay the output gives
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y1 n  x[Mn  no ]
yn - no   xMn  no 
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Causal System
• Causality
– A system is causal it’s output is a function of only the current and
previous samples
• Examples
– Backward Difference
y[n]  x[n]  x[n  1]
• Counter Example
– Forward Difference
y[n]  x[n  1]  x[n]
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Stable System
• Stability (in the sense of bounded-input bounded-output BIBO)
– A system is stable if and only if every bounded input produces a
bounded output
x[n]  Bx    y[n]  By  
• Example
– Square
y[n]  x[n]
2
if input is boundedby x[n]  Bx  
output is boundedby y[n]  B2x  
• Counter Example
– Log
y[n]  log10  x[n] 
evenif input is boundedby x[n]  Bx  
output not boundedfor xn  0  y0  log10  xn   
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