discrete-time signals and systems quote of the day mathematics is the tool specially suited for...

Discrete-Time Signals and Systems

Quote of the DayMathematics is the tool specially suited for

dealing with abstract concepts of any kind and 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.

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 2

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

0 20 40 60 80 100-10

0

10

t (ms)

0 10 20 30 40 50-10

0

10

n (samples)



Basic Sequences and Operations• Delaying (Shifting) a sequence

• Unit sample (impulse) sequence

• Unit step sequence

• Exponential sequences

]nn[x]n[y o

0n1

0n0]n[

0n1

0n0]n[u

nA]n[x

-10 -5 0 5 100

0.5

1

1.5

-10 -5 0 5 100

0.5

1

1.5

-10 -5 0 5 100

0.5

1



Sinusoidal Sequences• Important class of sequences

• An exponential sequence with complex

• x[n] is a sum of weighted sinusoids• Different from continuous-time, discrete-time sinusoids

– Have ambiguity of 2k in frequency

– Are not necessary periodic with 2/o

ncosnx o

jj eAA and e o

nsinAjncosAnx

eAeeAAnx

o

n

o

n

njnnjnjn oo

ncosnk2cos oo

integer an is k2

N if only Nncosncoso

ooo



Demo

Rotating Phasors Demo



Discrete-Time Systems• Discrete-Time Sequence is a mathematical operation that

maps a given input sequence x[n] into an output sequence y[n]

• Example Discrete-Time Systems– Moving (Running) Average

– Maximum

– Ideal Delay System

]}n[x{T]n[y T{.}x[n] y[n]

]3n[x]2n[x]1n[x]n[x]n[y

]2n[x],1n[x],n[xmax]n[y

]nn[x]n[y o



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

– Sign

• Counter Example– Ideal Delay System

2]n[x]n[y

]n[xsign]n[y

]nn[x]n[y o



Linear Systems• Linear System: A system is linear if and only if

• Examples– Ideal Delay System

(scaling) ]n[xaT]n[axT

and

y)(additivit ]n[xT]n[xT]}n[x]n[x{T 2121

]nn[x]n[y o

]nn[ax]n[xaT

]nn[ax]n[axT

]nn[x]nn[x]n[xT]}n[x{T

]nn[x]nn[x]}n[x]n[x{T

o1

o1

o2o112

o2o121



Time-Invariant Systems• Time-Invariant (shift-invariant) Systems

– A time shift at the input causes corresponding time-shift at output

• Example– Square

• Counter Example– Compressor System

]nn[xT]nn[y]}n[x{T]n[y oo

2]n[x]n[y

2oo

2o1

]nn[xn-ny gives output the Delay

]nn[xny is output the input the Delay

]Mn[x]n[y

oo

o1

nnMxn-ny gives output the Delay

]nMn[xny is output the input the Delay



Causal System• Causality

– A system is causal it’s output is a function of only the current and previous samples

• Examples– Backward Difference

• Counter Example– Forward Difference

]n[x]1n[x]n[y

]1n[x]n[x]n[y



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

• Example– Square

• Counter Example– Log

yx B]n[y B]n[x

2]n[x]n[y

2x

x

B]n[y by bounded is output

B]n[x by bounded is input if

]n[xlog]n[y 10

nxlog0y 0nxfor bounded not output

B]n[x by bounded is input if even

10

x

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