5 Kundur Test1Summary Handouts

8/13/2019 5 Kundur Test1Summary Handouts

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Introduction to Digital Signal Processing

Professor Deepa Kundur

University of Toronto

Professor Deepa Kundur (University of Toronto)Introduction to Digital Signal Processing 1 / 58

Chapter 1: Introduction

Analog and Digital Signals

analog signal = continuous-time + continuous amplitude

digital signal = discrete-time + discrete amplitude

t

2

1 2 3-1-2-3 4

-2

-4

x(t)

t

1

2

1 2 3-1-2-3 0.5 1.5 2.5 4

0.5

-2

-4

x(t)

-1 10n

x[n]

-2-3 2 3

1

-1 10n

x[n]

-2- 3 2 3

2

1 1

continuous-time

discrete-time

c on ti nuo us a mpl itu de di scre te a mp lit ude



Analog and Digital Signals

analog system = analog input + analog output

digital system = digital input + digital output



Discrete-time Sinusoids

x(n) = A cos(n+) = A cos(2fn+), n Z

discrete-time signal (not digital),

A xa(t) Aand n Z

A = amplitude

= frequency in rad/sample

f = frequency in cycles/sample; note: = 2f

= phase in rad



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3/15


Uniqueness: Continuous-time

ForF1=F2,

A cos(2F1t+) =A cos(2F2t+)

except at discrete points in time.



Uniqueness: Continuous-time

F1=F2:



Uniqueness: Discrete-time

Letf1= f0+kwhere k Z,

x1(n) = A e

j(2f1n+)

= A ej(2(f0+k)n+)

= A ej(2f0n+) ej(2kn)

= x0(n) 1= x0(n)



0 1 3 4-1-3 5

2-2 6

0 1 3 4-1-3 5

2-2 6



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Harmonically Related Complex Exponentials

Harmonically related sk(t) = ejk0t =ej2kF0t,

(cts-time) k= 0, 1, 2, . . .

Scientific Designation Frequency (Hz) kfor F0 = 8.176

C-1 8.176 1

C0 16.352 2C1 32.703 4C2 65.406 8C3 130.813 16C4 261.626 32

... ...

C9 8372.018 1024




Scientific Designation Frequency (Hz) k forF0 = 8.176

C1 32.703 4C2 65.406 8C3 130.813 16

C4 (middle C) 261.626 32C5 523.251 64

C6 1046.502 128C7 2093.005 256C8 4186.009 512




What does the family of harmonically related sinusoids sk(t) have incommon?

Harmonically related sk(t) = ejk0t =ej2(kF0)t,

(cts-time) k= 0, 1, 2, . . .

fund. period: T0,k = 1

cyclic frequency=

1

kF0period: Tk = any integer multiple ofT0

common period: T = k T0,k= 1

F0




Discrete-time Case:

For periodicity, select f0= 1N

where N Z:

Harmonically related sk(n) = ej2kf0n =ej2kn/N,(dts-time) k= 0, 1, 2, . . .

There are only Ndistinct dst-time harmonics:sk(n), k= 0, 1, 2, . . . ,N 1.



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Analog-to-Digital Conversion

CoderQuantizer

x(n)x (t)a

x (n)q

Analog

signal

Discrete-time

signal

Quantized

signal

Digital

signal

A/D converter

0 1 0 1 1 . . .

Sampler




CoderQuantizer

x(n)x (t)a

x (n)q

Analog

signal

Discrete-time

signal

Quantized

signal

Digital

signal

A/D converter

0 1 0 1 1 . . .

Sampler

Sampling: conversion from cts-time to dst-time by taking samples at

discrete time instants

E.g., uniform sampling: x(n) = xa(nT)where T is the samplingperiod and n Z




CoderQuantizer

x(n)x (t)a

x (n)q

Analog

signal

Discrete-time

signal

Quantized

signal

Digital

signal

A/D converter

0 1 0 1 1 . . .

Sampler

Quantization: conversion from dst-time cts-valued signal to a dst-time

dst-valued signal

quantization error: eq(n) = xq(n) x(n)for all n Z




CoderQuantizer

x(n)x (t)a

x (n)q

Analog

signal

Discrete-time

signal

Quantized

signal

Digital

signal

A/D converter

0 1 0 1 1 . . .

Sampler

Coding: representation of each dst-value xq(n) by a

b-bit binary sequence

e.g., if for any n, xq(n) {0, 1, . . . , 6, 7}, then the coder mayuse the following mapping to code the quantized amplitude:



6/15



CoderQuantizer

x(n)x (t)a

x (n)q

Analog

signal

Discrete-time

signal

Quantized

signal

Digital

signal

A/D converter

0 1 0 1 1 . . .

Sampler

Example coder:

0 000 4 1001 001 5 1012 010 6 1103 011 7 111



Sampling Theorem

If thehighest frequencycontained in an analog signal xa(t) isFmax= Band the signal is sampled at a rate

Fs> 2Fmax= 2B

thenxa(t) can be exactly recovered from its sample values using theinterpolation function

g(t) =sin(2Bt)

2Bt

Note: FN= 2B= 2Fmaxis called theNyquist rate.



Sampling Theorem

Sampling Period = T = 1

Fs=

1

Sampling Frequency

Therefore, given the interpolation relation, xa(t) can be written as

xa(t) =

n= x

a(nT)g(t nT)

xa(t) =

n=

x(n) g(t nT)

wherexa(nT) = x(n); calledbandlimited interpolation.



Bandlimited Interpolation



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Digital-to-Analog Conversion

Common interpolation approaches: bandlimited interpolation,zero-order hold, linear interpolation, higher-order interpolationtechniques, e.g., using splines

In practice, cheap interpolation along with a smoothing filteris employed.




-T T0

t-2T 2T 3T

1

original/bandlimited

interpolated signal

zero-orderhold

-3T






-T T0

t-2T 2T 3T

1

-3T

linear

interpolation

original/bandlimited

interpolated signal




Chapter 2: Dst-Time Signals & Systems

Elementary Discrete-Time Signals

1. unit sample sequence (a.k.a. Kronecker delta function):

(n) =

1, for n = 00, for n = 0

2. unit step signal:

u(n) = 1, for n 00, for n < 03. unit ramp signal:

ur(n) =

n, forn 00, forn < 0

Note:

(n) = u(n) u(n 1)=ur(n+ 1) 2ur(n) +ur(n 1)

u(n) = ur(n+ 1) ur(n)



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Signal Symmetry

Even signal: x(n) = x(n)

Odd signal: x(n) = x(n)

-1 10n

x(n)

-2-3-4-5-6-7 2 3 4 5 6 7

1

2

n-1 10-2-3-4-5-6-7 2 3 4 5 6 7

1

2

n-1 10-2-3-4-5-6-7 2 3 4 5 6 7

1

2

x(n) x(n)



Signal Symmetry

Even signal component: xe(n) = 12[x(n) +x(n)]

Odd signal component: xo(n) = 1

2[x(n) x(n)]

Note: x(n) = xe(n) +xo(n)



Signal Symmetry

x(n)

n

-1

1

0.5

-0.5

1 2 3 40 6 7 85-1-2-3-4-5-6

x(-n)

n

-1

1

0.5

-0.5

1 2 3 40 6 7 85-1-2-3-4-5-6

(x(n)+x(-n))/2

n

-1

1

0.5

-0.5

1 2 3 40 6 7 85-1-2-3-4-5-6

(x(n)-x(-n))/2

n

-1

1

0.5

-0.5

1 2 3 40 6 7 85

-1-2

-3-4-5-6

even part

odd part



Simple Manipulation of Discrete-Time Signals

Transformation of independent variable: time shift: n n k, k Z

Question: what ifk Z? time scale: n n, Z

Question: what if Z?

Additional, multiplication and scaling: amplitude scaling: y(n) = Ax(n),


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Simple Manipulation of Discrete-Time Signals I

Findx(n) x(n+ 1).

x(n)

n

-1

2

-2

3

-3

1

2 2

-2

3

-1

1

1 2 3 40 6 7 8 95

15 20

10-1-2-3



Simple Manipulation of Discrete-Time Signals II



Simple Manipulation of Discrete-Time Signals IFindx( 3

2n+ 1).

n 3n2

+ 1 x( 3n2

+ 1)

19 0if 3n2

+ 1 is an integer; undefinedotherwise



Simple Manipulation of Discrete-Time SignalsGraph ofx( 32 n+ 1).

n

-1

2

-2

3

-3

1

3

1 2 3 40 6 7 8 95

1211

10

-1-2-3 13 14

1

2 2

-2

-1

This signal is undefined for values of n

that are not even integers and zero for

even integers not shown on this sketch.



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11/15


Static vs. Dynamic

Examples: memoryless or not?

y(n) = A x(n), A = 0

y(n) = A x(n) +B, A,B, = 0

y(n) = x(n) cos(25 (n 5))

y(n) = x(n) y(n) = x(n+ 1)

y(n) = 11x(n+2)

y(n) = e3x(n)

y(n) =n

k=x(k)

Ans: Y, Y, Y, N, N, N, Y, N



Discrete-Time Bounded Signals



The Convolution Sum

Recall:

x(n) =

k=

x(k)(n k)



The Convolution Sum

Let the response of a linear time-invariant (LTI) system to the unitsample input (n) be h(n).

(n) T h(n)

(n k)

T

h(n k)(n k)

T h(n k)

x(k) (n k) T x(k) h(n k)

k=

x(k)(n k) T

k=

x(k)h(n k)

x(n) T y(n)



12/15


The Convolution Sum

Therefore,

y(n) =

k=

x(k)h(n k) = x(n) h(n)

for any LTI system.



Finite vs. Infinite Impulse Response

Implementation:Two classes

Finite impulse response (FIR):

y(n) =M1

k=0

h(k)x(n

k) nonrecursive systemsInfinite impulse response (IIR):

y(n) =k=0

h(k)x(n k)

recursive systems



System Realization

General expression for Nth-order LCCDE:

N

k=0aky(nk) =

M

k=0bkx(nk) a0 1

Initial conditions: y(1),y(2),y(3), . . . ,y(N).

Need: (1)constant scale, (2)addition, (3)delayelements.



Building Block Elements

Constant multiplier:

Signal multiplier: +

Unit delay:

Unit advance:

Adder: +



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14/15

Chapter 3: The z-Transform and Its Applications

The Direct z-Transform

Direct z-Transform:

X(z) =

n=

x(n)zn

Notation:

X(z) Z{x(n)}

x(n) Z X(z)



Region of Convergence

the region of convergence (ROC) ofX(z) is the set of all valuesofz for which X(z) attains a finite value

Thez-Transform is, therefore, uniquely characterized by:

1. expression for X(z)2. ROC ofX(z)



ROC Families: Finite Duration Signals



ROC Families: Infinite Duration Signals



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5 Kundur Test1Summary Handouts