dsp slides module2 0

Digital Signal Processing

Module 2: Discrete-time signalsDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Video Introduction

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Module Overview:

Module 2.1: discrete-time signals and operators

Module 2.2: the discrete-time complex exponential

Module 2.3: the Karplus-Strong algorithm

2 1

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


Module 2.1: Discrete-time signalsDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Overview:

discrete-time signals

signal classes

elementary operators

shifts

energy and power

2.1 2

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Discrete-time signals have a long tradition...

Meteorology (limnology): the floods of the Nile

Representations of flood data: circa 2500 BC

2.1 3

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Discrete-time signals have a long tradition...

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1500

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4500

1865 1890 1915 1940 1965 1990

year

m3/s

Representations of flood data: circa AD 2000

2.1 4

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Probably your first scientific experiment...

Daily temperature

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C

2.1 5

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Astronomy

monthly solar spot activity, 1749 to 2003

1749 1875 20000

100

200

month

2.1 6

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

History and sociology

world population (billions)

1AD 1700AD 2030AD0

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year

2.1 7

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

Economics

a purely man-made signal: the Dow Jones industrial average

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1891 1916 1941 1966 1991 2016

year

2.1 8

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

More formally...

Discrete-time signal: a sequence of complex numbers

one dimension (for now)

notation: x [n]

two-sided sequences: x : Z C

n is dimension-less time

analysis: periodic measurement

synthesis: stream of generated samples

2.1 9

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

The delta signal

x [n] = [n]

b b b b b b b b b b b b b b b b

b


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2.1 10

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

How do you synchronize audio and video...

2.1 11

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How do you synchronize audio and video...

2.1 12

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

The unit step

x [n] = u[n]


b b b b b b b b b b b b b b b b b

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2.1 13

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

The Frankenstein switch...

2.1 14

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

The exponential decay

x [n] = |a|n u[n], |a| < 1


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2.1 15

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

How fast does your coffee get cold...

2.1 16

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

How fast does your coffee get cold...

Newtons law of cooling:dT

dt= c(T Tenv)

T (t) = Tenv + (T0 Tenv)ect

In practice:

must have convection only

must have large conductivity

2.1 17

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

The sinusoid

x [n] = sin(0n + )

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2.1 18

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Oscillations are everywhere!

2.1 19

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Four signal classes

finite-length

infinite-length

periodic

finite-support

2.1 20

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Finite-length signals

sequence notation: x [n], n = 0, 1, . . . ,N 1

vector notation: x = [x0 x1 . . . xN1]T

practical entities, good for numerical packages (Matlab and the like)

2.1 21

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Infinite-length signals

sequence notation: x [n], n Z

abstraction, good for theorems

2.1 22

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Periodic signals

N-periodic sequence: x [n] = x [n + kN], n, k ,N Z

same information as finite-length of length N

natural bridge between finite and infinite lengths

2.1 23

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Finite-support signals

Finite-support sequence:

x [n] =

x [n] if 0 n < N

0 otherwise

n Z

same information as finite-length of length N

another bridge between finite and infinite lengths

2.1 24

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Elementary operators

scaling:y [n] = x [n]

sum:y [n] = x [n] + z [n]

product:y [n] = x [n] z [n]

shift by k (delay):y [n] = x [n k]

2.1 25

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Shift of a finite-length: finite-support

[x0 x1 x2 x3 x4 x5 x6 x7]

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0 1 2 3 4 5 6 70

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2.1 26

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n]

. . . x0 x1 x2 x3 x4 x5 x6 x7 . . .

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2.1 26

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n]

. . . 0 0 0 x0 x1 x2 x3 x4 x5 x6 x7 0 0 0 . . .

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2.1 26

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n 1]

. . . 0 0 0 0 x0 x1 x2 x3 x4 x5 x6 x7 0 0 . . .

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2.1 26

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n 2]

. . . 0 0 0 0 0 x0 x1 x2 x3 x4 x5 x6 x7 0 . . .

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2.1 26

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n 3]

. . . 0 0 0 0 0 0 x0 x1 x2 x3 x4 x5 x6 x7 . . .

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2.1 26

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n 4]

. . . 0 0 0 0 0 0 0 x0 x1 x2 x3 x4 x5 x6 . . .

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2.1 26

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Shift of a finite-length: periodic extension

[x0 x1 x2 x3 x4 x5 x6 x7]

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2.1 27

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n]

. . . x0 x1 x2 x3 x4 x5 x6 x7 . . .

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2.1 27

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n]

. . . x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 x7 x0 x1 x2 . . .

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2.1 27

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n 1]

. . . x4 x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 x7 x0 x1 . . .

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2.1 27

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n 2]

. . . x3 x4 x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 x7 x0 . . .

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2.1 27

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n 3]

. . . x2 x3 x4 x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 x7 . . .

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2.1 27

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n 4]

. . . x1 x2 x3 x4 x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 . . .

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2.1 27

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Energy and power

Ex =

n=

|x [n]|2

Px = limN

1

2N + 1

Nn=N

|x [n]|2

2.1 28

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Energy and power: periodic signals

Ex =

Px 1

N

N1n=0

|x [n]|2

2.1 29

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

END OF MODULE 2.1

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


Module 2.2: the complex exponentialDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Overview:

the complex exponential

periodicity

wagonwheel effect and maximum speed

digital and real-world frequency

2.2 30

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Oscillations are everywhere

2.2 31

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

The oscillatory heartbeat

Ingredients:

a frequency (units: radians)

an initial phase (units: radians)

an amplitude A (units depending on underlying measurement)

a trigonometric function

e.g. x [n] = A cos(n + )

2.2 32

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

The complex exponential

the trigonometric function of choice in DSP is the complex exponential:

x [n] = Ae j(n+)

= A[cos(n + ) + j sin(n + )]

2.2 33

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Why complex exponentials?

makes sense: sines and cosines always go together

simpler math: trigonometry becomes algebra

we can use complex numbers in digital systems

2.2 34

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

The advantages of complex exponentials

Example: change the phase of a pure cosine

cos(n + ) = a cos(n) + b sin(n), a = cos, b = sin

each sinusoid is always a sum of sine and cosine

we have to remember complex trigonometric formulas

we have to carry more terms in our equations

2.2 35

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

The advantages of complex exponentials

Example: change the phase of a pure cosine

Re{e j(n+)} = Re{e jn e j}

sine and cosine live together

phase shift is simple multiplication

notation is simpler

2.2 36

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013


e j = cos+ j sin

1 1

1

1

Re

Im

ej

b

2.2 37

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013


z: point on the complex plane

Re

Im

zb

2.2 38

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013


rotation: z = z e j

Re

Im

zb

zb

zb

2.2 38

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

The complex exponential generating machine

x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

x[0]b

2.2 39

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

x[1]b

2.2 39

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

x[2]b

2.2 39

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

x[3]b

2.2 39

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

x[4]b

2.2 39

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

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2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

x[5]b

2.2 39

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

x[6] b

2.2 39

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

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2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

x[7]b

2.2 39

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

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2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

x[8]

b

2.2 39

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

x[9]

b

2.2 39

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

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2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

x[10]

b

2.2 39

Digital Sig

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Paolo Pran

doni and M

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x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

b

x[11]b

2.2 39

Digital Sig

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Paolo Pran

doni and M

artin Vett

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x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

b

b

x[12]b

2.2 39

Digital Sig

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Paolo Pran

doni and M

artin Vett

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x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

b

b

b

x[13]b

2.2 39

Digital Sig

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Paolo Pran

doni and M

artin Vett

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x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

b

b

b

b

x[14]b

2.2 39

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

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2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

x[0]b

2.2 40

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

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2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

x[1]b

2.2 40

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

x[2]b

2.2 40

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

x[3]b

2.2 40

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

x[4]b

2.2 40

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

x[5]b

2.2 40

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

b

x[6] b

2.2 40

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

b

b

x[7]b

2.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

b

b

b

x[8]

b

2.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

b

b

b

b

x[9]

b

2.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

b

b

b

b

b

x[10]

b

2.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

b

b

b

b

b

b

x[11]b

2.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

b

b

b

b

b

b

b

x[12]b

2.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

b

b

b

b

b

b

b

b

x[13]b

2.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Initial phase

x [n] = e j(n+); x [n + 1] = e jx [n], x [0] = e j

Re

Im

b

b

b

b

b

b

b

b

b

b

b

b

b

b

x[14]b

2.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Careful: not every sinusoid is periodic in discrete time

x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

x[0]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

x[1]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

x[2]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

x[3]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

x[4] b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

x[5]

b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

x[6]

b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

x[7]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

x[8]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

x[9]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

x[10]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

b

x[11]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

b

b

x[12]b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

b

b

b

x[13]

b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] = e jn; x [n + 1] = e jx [n]

Re

Im

b

b

b

b

b

b

b

b

b

b

b

b

b

b

x[14]

b

2.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Periodicity

e jn periodic =M

N2pi,M,N N

e j = e j(+2kpi) k N

2.2 42

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

One point, many names

Re

Im

ej

b

2.2 43

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


Re

Im

ej

b

2pi +

2.2 43

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


Re

Im

ej

b

6pi +

2.2 43

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


Re

Im

ej

b

2.2 44

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


Re

Im

ej

b

2pi +

2.2 44

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

2.2 45

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/12

Re

Im

x[0]b

x[1]b

x[2]b

x[3]b

x[4]b

x[5]b

x[6] b

x[7]b

x[8]

b

x[9]

b

x[10]

b

x[11]b

2.2 46

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/6

Re

Im

x[0]b

x[1]b

x[2]b

x[3] b

x[4]

b

x[5]

b

2.2 47

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/5

Re

Im

x[0]b

x[1]b

x[2]b

x[3]b

x[4]

b

2.2 48

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/4

Re

Im

x[0]b

x[1]b

x[2] b

x[3]

b

2.2 49

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/3

Re

Im

x[0]b

x[1]b

x[2]

b

2.2 50

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/2 = pi

Re

Im

x[0]b

2.2 51

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/2 = pi

Re

Im

x[0] b

2.2 51

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/2 = pi

Re

Im

x[1]b

2.2 51

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/2 = pi

Re

Im

x[2] b

2.2 51

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/2 = pi

Re

Im

x[3]b

2.2 51

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/2 = pi

Re

Im

x[4] b

2.2 51

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How fast can we go?

= 2pi/2 = pi

Re

Im

x[5]b

2.2 51

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

What if we go faster?

pi < < 2pi

Re

Im

x[0]b

x[1]b

2.2 52

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

What if we go faster?

pi < < 2pi

Re

Im

x[0]b

x[1]b

2pi

2.2 52

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Lets go really too fast

= 2pi , small

Re

Im

x[0]b

2.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


= 2pi , small

Re

Im

b

x[1]b

2.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


= 2pi , small

Re

Im

b

b

x[2]b

2.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


= 2pi , small

Re

Im

b

b

b

x[3]b

2.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


= 2pi , small

Re

Im

b

b

b

b

x[4]b

2.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


= 2pi , small

Re

Im

b

b

b

b

b

x[5]

b

2.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


= 2pi , small

Re

Im

b

b

b

b

b

b

x[6]

b

2.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


= 2pi , small

Re

Im

b

b

b

b

b

b

b

x[7]

b

2.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

The wagonwheel effect

2.2 54

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Digital vs physical frequency

Discrete time:

n: no physical dimension (just a counter)

periodicity: how many samples before pattern repeats

Real world:

periodicity: how many seconds before pattern repeats

frequency measured in Hz (s1)

2.2 55

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How your PC plays sounds

x [n] sound card

2.2 56

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How your PC plays sounds

x [n] sound card

Ts system clock

2.2 56

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Digital vs physical frequency

set Ts , time in seconds between samples

periodicity of M samples periodicity of MTs seconds

real world frequency:

f =1

MTs

2.2 57

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

END OF MODULE 2.2

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


Module 2.3: the Karplus-Strong algorithmDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Overview:

DSP building blocks

moving averages and simple feedback loops

a sound synthesizer

2.3 58

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Overview:

DSP as Lego: The fundamental building blocks

Averages and moving averages

Recursion: Revisiting your bank account

Building a simple recursive synthesizer

Examples of sounds

2.3 59

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

DSP as Lego

x [n] b + b + y [n]

z1

+ b z3

z1

a b

1

c

2.3 60

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Building Blocks: Adder

x [n]

+ x [n] + y [n]

y [n]

2.3 61

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n]

+ x [n] + y [n]

y [n]

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

2.3 61

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n]

+ x [n] + y [n]

y [n]

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

b b b b b b b b b b b

0 2 4 6 8 100

1

2.3 61

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Building Blocks: Multiplier

x [n] x [n]

2.3 62

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] x [n]

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

2.3 62

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] x [n]

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

= 0.5

2.3 62

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Building Blocks: Unit Delay

x [n] z1 x [n 1]

2.3 63

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] z1 x [n 1]

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

2.3 63

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] z1 x [n 1]

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

bb

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

2.3 63

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Building Blocks: Arbitrary Delay

x [n] zN x [n N]

2.3 64

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] zN x [n N]

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

2.3 64

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013


x [n] zN x [n N]

b

b

b

b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

b b b b

b

b

b

b

b

b

b

0 2 4 6 8 100

1

N = 4

2.3 64

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

The 2-point Moving Average

simple average:

m =a + b

2

moving average: take a local average

y [n] =x [n] + x [n 1]

2

2.3 65

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

The 2-point Moving Average Using Lego

x [n] b + y [n]

z1

1/2

2.3 66

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Lets average...

x [n] = [n]

b b

b

b b b b b b b b b b

2 0 2 4 6 8 10

1

0

1

2.3 67

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Lets average...

x [n] = [n]

b b

b

b b b b b b b b b b

2 0 2 4 6 8 10

1

0

1

b b

b b

b b b b b b b b b

2 0 2 4 6 8 10

1

0

1

2.3 67

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Lets average...

x [n] = u[n]

b b


2 0 2 4 6 8 10

1

0

1

2.3 68

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Lets average...

x [n] = u[n]

b b


2 0 2 4 6 8 10

1

0

1

b b

b

b b b b b b b b b b

2 0 2 4 6 8 10

1

0

1

2.3 68

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Lets average...

x [n] = cos(n), = pi/10

b

b

b

b

b

b

b

b

b

b

b

b

b

2 0 2 4 6 8 10

1

0

1

2.3 69

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Lets average...

x [n] = cos(n), = pi/10

b

b

b

b

b

b

b

b

b

b

b

b

b

2 0 2 4 6 8 10

1

0

1b

b

b b

b

b

b

b

b

b

b

b

b

2 0 2 4 6 8 10

1

0

1

2.3 69

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Lets average...

x [n] = cos(n), = pi

b

b

b

b

b

b

b

b

b

b

b

b

b

2 0 2 4 6 8 10

1

0

1

2.3 70

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Lets average...

x [n] = cos(n), = pi

b

b

b

b

b

b

b

b

b

b

b

b

b

2 0 2 4 6 8 10

1

0

1

b b b b b b b b b b b b b

2 0 2 4 6 8 10

1

0

1

2.3 70

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

What if we reverse the loop?

x [n] b + y [n]

z1

1/2

2.3 71

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

What if we reverse the loop?

x [n] + b y [n]

z1

2.3 71

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

A simple model for banking

A simple equation to describe compound interest:

constant interest/borrowing rate of 5% per year

interest accrues on Dec 31

deposits/withdrawals during year n: x [n]

balance at year n:y [n] = 1.05 y [n 1] + x [n]

2.3 72

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

First-order recursion

x [n] + b y [n]

z11.05

y [n] = 1.05 y [n 1] + x [n]

2.3 73

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Example: the one-time investment

x [n] = 100 [n]

y [0] = 100

y [1] = 105

y [2] = 110.25, y [3] = 115.7625 etc.

In general: y [n] = (1.05)n100 u[n]

b b

b

b b b b b b b b b b

2 0 2 4 6 8 100

100

200

2.3 74

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Example: the one-time investment

x [n] = 100 [n]

y [0] = 100

y [1] = 105

y [2] = 110.25, y [3] = 115.7625 etc.

In general: y [n] = (1.05)n100 u[n]

b b

b

b b b b b b b b b b

2 0 2 4 6 8 100

100

200

b b

b

b

b

b

b

b

b

b

b

b

b

2 0 2 4 6 8 100

100

200

2.3 74

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Example: the saver

x [n] = 100 u[n]

y [0] = 100

y [1] = 205

y [2] = 315.25, y [3] = 431.0125 etc.

In general: y [n] = 2000 ((1.05)n+1 1) u[n]

b b


2 0 2 4 6 8 100

100

200

2.3 75

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Example: the saver

x [n] = 100 u[n]

y [0] = 100

y [1] = 205

y [2] = 315.25, y [3] = 431.0125 etc.

In general: y [n] = 2000 ((1.05)n+1 1) u[n]

b b


2 0 2 4 6 8 100

100

200

b b

b

b

b

b

b

b

b

b

b

b

b

2 0 2 4 6 8 100

500

1000

1500

2.3 75

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Example: The independently wealthy

x [n] = 100 [n] 5 u[n 1]

y [0] = 100

y [1] = 100

y [2] = 100, y [3] = 100 etc.

In general: y [n] = 100 u[n]

b b

b

b b b b b b b b b b

2 0 2 4 6 8 10

0

100

2.3 76

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Example: The independently wealthy

x [n] = 100 [n] 5 u[n 1]

y [0] = 100

y [1] = 100

y [2] = 100, y [3] = 100 etc.

In general: y [n] = 100 u[n]

b b

b

b b b b b b b b b b

2 0 2 4 6 8 10

0

100

b b


2 0 2 4 6 8 100

100

200

2.3 76

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

A simple generalization

x [n] + b y [n]

zM

y [n] = y [n M] + x [n]

2.3 77

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Example

M = 3, = 0.7, x [n] = [n]

y [0] = 1, y [1] = 0, y [2] = 0

y [3] = 0.7, y [4] = 0, y [5] = 0

y [6] = 0.72, y [7] = 0, y [8] = 0, etc.

b b

b

b b b b b b b b b b

2 0 2 4 6 8 100

1

2.3 78

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Example

M = 3, = 0.7, x [n] = [n]

y [0] = 1, y [1] = 0, y [2] = 0

y [3] = 0.7, y [4] = 0, y [5] = 0

y [6] = 0.72, y [7] = 0, y [8] = 0, etc.

b b

b

b b b b b b b b b b

2 0 2 4 6 8 100

1

b b

b

b b

b

b b

b

b b

b

b

2 0 2 4 6 8 100

1

2.3 78

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Example

M = 3, = 1, x [n] = [n] + 2 [n 1] + 3 [n 2]

y [0] = 1, y [1] = 2, y [2] = 3

y [3] = 1, y [4] = 2, y [5] = 3

y [6] = 1, y [7] = 2, y [8] = 3, etc.

b b

b

b

b

b b b b b b b b

2 0 2 4 6 8 100

1

2

3

b b

b

b

b

b

b

b

b

b

b

b

b

2 0 2 4 6 8 100

1

2

3

2.3 79

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

We can make music with that!

build a recursion loop with a delay of M

choose a signal x [n] that is nonzero only for 0 n < M

choose a decay factor

input x [n] to the system

play the output

2.3 80

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

How do we play it, really?

M-tap delay M-sample periodicity

associate time T to sample interval

periodic signal of frequency

f =1

MTHz

example: T = 22.7s, M = 100f 440Hz

2.3 81

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Playing a sine wave

M = 100, = 1, x [n] = sin(2pi n/100) for 0 n < 100 and zero elsewhere

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0 16 32 48 64 80 96

1

0

1

2.3 82

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Playing a sine wave

M = 100, = 1, x [n] = sin(2pi n/100) for 0 n < 100 and zero elsewhere

b

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0 16 32 48 64 80 96

1

0

1

0 166 332 498 664 830 996

1

0

1

2.3 82

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Introducing some realism

M controls frequency (pitch)

controls envelope (decay)

x [n] controls color (timbre)

2.3 83

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

A proto-violin

M = 100, = 0.95, x [n]: zero-mean sawtooth wave between 0 and 99, zero elsewhere

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0 16 32 48 64 80 96

1

0

1

2.3 84

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

A proto-violin

M = 100, = 0.95, x [n]: zero-mean sawtooth wave between 0 and 99, zero elsewhere

b

b

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0 16 32 48 64 80 96

1

0

1

0 166 332 498 664 830 996

1

0

1

2.3 84

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

The Karplus-Strong Algorithm

M = 100, = 0.9, x [n]: 100 random values between 0 and 99, zero elsewhere

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0 16 32 48 64 80 96

1

0

1

2.3 85

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

The Karplus-Strong Algorithm

M = 100, = 0.9, x [n]: 100 random values between 0 and 99, zero elsewhere

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