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Digital Signal Processing

Module 2: Discrete-time signalsDigital Signal Processing

Paolo Prandoni and Martin Vetterli

© 2013

Video Introduction



© 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



© 2013


Module 2.1: Discrete-time signalsDigital Signal Processing


© 2013

Overview:

◮ discrete-time signals

◮ signal classes

◮ elementary operators

◮ shifts

◮ energy and power

2.1 2



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Discrete-time signals

Economics: the Dow Jones industrial average

0

5000

10000

1891 1916 1941 1966 1991 2016

year

2.1 3



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Astronomy: monthly solar spot activity, 1749 to 2003

1749 1875 20000

100

200

month

2.1 4



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History: world population (billions)

1AD 1700AD 2030AD0

1

2

3

4

5

6

7

8

9

year

2.1 5



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Discrete-time signals have a long tradition...

Meteorology (limnology): the floods of the Nile

Representations of flood data: circa 2500 BC and today

2.1 6



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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 7



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The delta signal

x [n] = δ[n]

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

b


−15 −10 −5 0 5 10 150

1

2.1 8



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The unit step

x [n] = u[n]


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

−15 −10 −5 0 5 10 150

1

2.1 9



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The exponential decay

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


b

b

bbbbb b b b b b b b b b b

−15 −10 −5 0 5 10 150

1

2.1 10



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The sinusoid

x [n] = sin(ω0n + θ)

b

b

b

b

bb b b

b

b

b

b

b

b

bb b b

b

b

b

b

b

b

bb b b

b

b

b

b

b

−15 −10 −5 0 5 10 15

−1

0

1

2.1 11



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Four signal classes

◮ finite-length

◮ infinite-length

◮ periodic

◮ finite-support

2.1 12



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Finite-length signals

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

◮ vector notation: x = [x0 x1 . . . xN−1]T

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

2.1 13



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Infinite-length signals

◮ sequence notation: x [n], n ∈ Z

◮ abstraction, good for theorems

2.1 14



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Periodic signals

◮ N-periodic sequence: x̃ [n] = x̃ [n + kN], n, k ,N ∈ Z

◮ same information as finite-length of length N

◮ “natural” bridge between finite and infinite lengths

2.1 15



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Finite-support signals

◮ Finite-support sequence:

x̄ [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 16



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Elementary operators

◮ scaling:y [n] = αx [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 17



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Shift of a finite-length: finite-support

[x0 x1 x2 x3 x4 x5 x6 x7]

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

0 1 2 3 4 5 6 70

1

2.1 18



© 2013


x [n]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1 bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

2.1 18



© 2013


x̄ [n]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1 bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

2.1 18



© 2013


x̄ [n − 1]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

b

bb

bb

bb

b

0 1 2 3 4 5 6 70

1

2.1 18



© 2013


x̄ [n − 2]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

b b

bb

bb

bb

0 1 2 3 4 5 6 70

1

2.1 18



© 2013


x̄ [n − 3]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

b b b

bb

bb

b

0 1 2 3 4 5 6 70

1

2.1 18



© 2013


x̄ [n − 4]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

b b b b

bb

bb

0 1 2 3 4 5 6 70

1

2.1 18



© 2013

Shift of a finite-length: periodic extension

[x0 x1 x2 x3 x4 x5 x6 x7]

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

0 1 2 3 4 5 6 70

1

2.1 19



© 2013


x [n]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1 bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

2.1 19



© 2013


x̃ [n]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1 bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

2.1 19



© 2013


x̃ [n − 1]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

b

bb

bb

bb

b

0 1 2 3 4 5 6 70

1

2.1 19



© 2013


x̃ [n − 2]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

2.1 19



© 2013


x̃ [n − 3]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

bb

b

bb

bb

b

0 1 2 3 4 5 6 70

1

2.1 19



© 2013


x̃ [n − 4]

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

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

bb

bb

bb

bb

0 1 2 3 4 5 6 70

1

2.1 19



© 2013

Energy and power

Ex =∞∑

n=−∞

|x [n]|2

Px = limN→∞

1

2N + 1

N∑

n=−N

|x [n]|2

2.1 20



© 2013

Energy and power: periodic signals

Ex̃ = ∞

Px̃ ≡1

N

N−1∑

n=0

|x̃ [n]|2

2.1 21



© 2013

END OF MODULE 2.1



© 2013


Module 2.2: the complex exponentialDigital Signal Processing


© 2013

Overview:

◮ the complex exponential

◮ periodicity

◮ wagonwheel effect and maximum “speed”

◮ digital and real-world frequency

2.2 22



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The complex exponential

The most important discrete-time signal in the world:

x [n] = e j(ωn+φ)

2.2 23



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Recall: e jα = cosα+ j sinα

−1 1

−1

1

Re

Im

ejα

b

α

2.2 24



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Rotation factor: z′ = z e jα

Re

Im

zb

2.2 25



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Rotation factor: z′ = z e jα

Re

Im

zb

z′b

α

2.2 25



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The complex exponential generating machine

x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

x[0]b

2.2 26



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x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

x[1]b

ω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

x[2]b

ω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

x[3]b

ω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bbx[4]

bω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

x[5] b ω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

b

x[6] bω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

b

b

x[7]b

ω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

b

b

b

x[8]

b

ω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

b

b

b

b

x[9]

b

ω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

b

b

b

bb

x[10]

bω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

b

b

b

bb

b

x[11]bω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

b

b

b

bb

b

b

x[12]bω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

b

b

b

bb

b

b

b

x[13]b

ω

2.2 26



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

bb

b

b

b

b

bb

b

b

b

b

x[14]b

ω

2.2 26



© 2013

Initial phase

x [n] = e j(ωn+φ); x [n + 1] = e jωx [n], x [0] = e jφ

Re

Im

x[0]b

2.2 27



© 2013

Initial phase


Re

Im

b

x[1]b

ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

x[2]b

ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bx[3]

b

ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbx[4]

bω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

x[5] b ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

b

x[6] bω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

b

b

x[7]b

ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

b

b

b

x[8]

b

ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

b

b

b

b

x[9]

b

ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

b

b

b

b b x[10]

bω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

b

b

b

b bb

x[11]b

ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

b

b

b

b bb

b

x[12]b

ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

b

b

b

b bb

b

b

x[13]b

ω

2.2 27



© 2013

Initial phase


Re

Im

b

b

bbb

b

b

b

b bb

b

b

b

x[14]b

ω

2.2 27



© 2013

Careful: not every sinusoid is periodic in discrete time

x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

x[0]b

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

x[1]b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

x[2]b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

x[3]b ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

x[4] b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

x[5]

b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

b

x[6]

b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

bb

x[7]b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

bb

b

x[8]b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

bb

b

b

x[9]b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

bb

b

b

bx[10]

b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

bb

b

b

bb

x[11] bω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

bb

b

b

bb

b

x[12]b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

bb

b

b

bb

b

b

x[13]

b

ω

2.2 28



© 2013


x [n] = e jωn; x [n + 1] = e jωx [n]

Re

Im

b

b

b

b

b

bb

b

b

bb

b

b

bx[14]

bω

2.2 28



© 2013

Periodicity

e jωn periodic ⇐⇒ ω =M

N2π,M,N ∈ N

e jω = e j(ω+2kπ) ∀k ∈ N

2.2 29



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Quiz

◮ is the signal e jn periodic?

2.2 30



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One point, many names

Re

Im

ejα

b

α

2.2 31



© 2013


Re

Im

ejα

b

2π + α

2.2 31



© 2013


Re

Im

ejα

b

6π + α

2.2 31



© 2013


Re

Im

ejα

b

α

2.2 32



© 2013


Re

Im

ejα

b

α

−2π + α

2.2 32



© 2013

How “fast” can we go?

2.2 33



© 2013


ω = 2π/12

Re

Im

x[0]b

x[1]b

x[2]b

x[3]bx[4]

b

x[5] b

x[6] b

x[7]b

x[8]

b

x[9]

bx[10]

b

x[11]b

2.2 34



© 2013


ω = 2π/6

Re

Im

x[0]b

x[1]b

x[2]b

x[3] b

x[4]

b

x[5]

b

2.2 35



© 2013


ω = 2π/5

Re

Im

x[0]b

x[1]b

x[2] b

x[3]b

x[4]

b

2.2 36



© 2013


ω = 2π/4

Re

Im

x[0]b

x[1]b

x[2] b

x[3]

b

2.2 37



© 2013


ω = 2π/3

Re

Im

x[0]b

x[1]b

x[2]

b

2.2 38



© 2013


ω = 2π/2 = π

Re

Im

x[0]b

2.2 39



© 2013


ω = 2π/2 = π

Re

Im

x[0] b

2.2 39



© 2013


ω = 2π/2 = π

Re

Im

x[1]b

2.2 39



© 2013


ω = 2π/2 = π

Re

Im

x[2] b

2.2 39



© 2013


ω = 2π/2 = π

Re

Im

x[3]b

2.2 39



© 2013


ω = 2π/2 = π

Re

Im

x[4] b

2.2 39



© 2013


ω = 2π/2 = π

Re

Im

x[5]b

2.2 39



© 2013

What if we go “faster”?

π < ω < 2π

Re

Im

x[0]b

x[1]b

2.2 40



© 2013

What if we go “faster”?

π < ω < 2π

Re

Im

x[0]b

x[1]b

ω − 2π

2.2 40



© 2013

Let’s go really too fast

ω = 2π − α, α small

Re

Im

x[0]b

2.2 41



© 2013



Re

Im

b

x[1]bω

2.2 41



© 2013



Re

Im

bb

x[2]b

ω

2.2 41



© 2013



Re

Im

bbb

x[3]b

ω

2.2 41



© 2013



Re

Im

bbbb

x[4]b

ω

2.2 41



© 2013



Re

Im

bbbb

b

x[5]

b

ω

2.2 41



© 2013



Re

Im

bbbb

bb

x[6]

b

ω

2.2 41



© 2013



Re

Im

bbbb

bb

b

x[7]

b

ω

2.2 41



© 2013

The wagonwheel effect

2.2 42



© 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 (s−1)

2.2 43



© 2013

How your PC plays sounds

x [n] sound card

2.2 44



© 2013

How your PC plays sounds

x [n] sound card

Ts system clock

2.2 44



© 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 45



© 2013

END OF MODULE 2.2



© 2013


Module 2.3: the Karplus-Strong algorithmDigital Signal Processing


© 2013

Overview:

◮ DSP building blocks

◮ moving averages and simple feedback loops

◮ a sound synthesizer

2.3 46



© 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 47



© 2013

DSP as Lego

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

z−1

+ b z−3

z−1

a b

−1

c

2.3 48



© 2013

Building Blocks: Adder

x [n]

+ x [n] + y [n]

y [n]

2.3 49



© 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


0 2 4 6 8 100

1

2.3 49



© 2013


x [n]

+ x [n] + y [n]

y [n]


0 2 4 6 8 100

1


0 2 4 6 8 100

1


0 2 4 6 8 100

1

2.3 49



© 2013

Building Blocks: Multiplier

x [n] αx [n]α

2.3 50



© 2013


x [n] αx [n]α

bb

bb

bb

bb

bb

b

0 2 4 6 8 100

1

2.3 50



© 2013


x [n] αx [n]α

bb

bb

bb

bb

bb

b

0 2 4 6 8 100

1


0 2 4 6 8 100

1

α = 0.5

2.3 50



© 2013

Building Blocks: Unit Delay

x [n] z−1 x [n − 1]

2.3 51



© 2013


x [n] z−N x [n − N]

bb

bb

bb

bb

bb

b

0 2 4 6 8 100

1

b b b b

bb

bb

bb

b

0 2 4 6 8 100

1

N = 4

2.3 52



© 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 53



© 2013

Let’s average...

x [n] = δ[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 55



© 2013

Let’s 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 56



© 2013

Let’s average...

x [n] = cos(ωn), ω = π/10

bb b b

b

b

b

b

b

b

bb b

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Let’s average...

x [n] = cos(ωn), ω = π

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Let’s average...

x [n] = cos(ωn), ω = π

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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]

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© 2013

Example: the one-time investment

x [n] = 100 δ[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]

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© 2013

Example: the one-time investment

x [n] = 100 δ[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]

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© 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]

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© 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]

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© 2013

Example: The independently wealthy

x [n] = 100 δ[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]

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© 2013

Example: The independently wealthy

x [n] = 100 δ[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]

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Example

M = 3, α = 0.7, x [n] = δ[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.

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© 2013

Example

M = 3, α = 0.7, x [n] = δ[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.

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Example

M = 3, α = 1, x [n] = δ[n] + 2 δ[n − 1] + 3 δ[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.

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We can make music with that!

◮ build a recursion loop with a delay of M

◮ choose a signal x̄ [n] that is nonzero only for 0 ≤ n < M

◮ choose a decay factor

◮ input x̄ [n] to the system

◮ play the output

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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 = 22.7µs, M = 100f ≈ 440Hz

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© 2013

Playing a sine wave

M = 100, α = 1, x̄ [n] = sin(2π n/100) for 0 ≤ n < 100 and zero elsewhere

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© 2013

Playing a sine wave

M = 100, α = 1, x̄ [n] = sin(2π n/100) for 0 ≤ n < 100 and zero elsewhere

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Introducing some realism

◮ M controls frequency (pitch)

◮ α controls envelope (decay)

◮ x̄ [n] controls color (timbre)

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A proto-violin

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

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© 2013

A proto-violin

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

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The Karplus-Strong Algorithm

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

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© 2013

The Karplus-Strong Algorithm

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

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Recap

◮ We have seen basic elements:

• adders

• multipliers

• delays

◮ We have seen two systems

• moving averages

• recursive systems

◮ We were able to build simple systems with interesting properties

◮ to understand all of this in more details we need a mathematical framework!

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© 2013

w



© 2013

Exercises (2.1)

Q1 Sketch the signal u[n − 1]

Q2 Sketch the signal u[n]− u[n − 4]

Q3 Prove that

x [n] =∞∑

k=−∞

x [k]δ[n − k]

Q4 Prove that

u[n] =

n∑

k=−∞

δ[k]

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© 2013

Answers: Q2

x [n] = u[n]− u[n − 4]



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© 2013

Answers: Q3

x [n] =∞∑

k=−∞

x [k]δ[n − k]

δ[n − k] =

1 if n = k

0 otherwise

n ∈ Z

Therefore, the only term in the sum that is not “killed” is the one for k = n, i.e. x [n].

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© 2013

Answers: Q4

u[n] =

n∑

k=−∞

δ[k]

Call y [n] =∑n

k=−∞δ[k].

If n < 0 all the terms in the sum are zero, so y [n] = 0 for n < 0If n > 0, the sum will have just one nonzero term for k = 0, so y [n] = 1 for n ≥ 0.Therefore y [n] = u[n]

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© 2013

Quiz (2.2)

What is the minimum period P (in samples) of the signal e j(M/N)2πn for the following valuesof M,N?

Q1 M = 1,N = 3?

Q2 M = 5,N = 7?

Q3 M = 35,N = 15?

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© 2013

Answer

n = 0

x[n] = . . . 0, 0, 1, 2, 3, 0, 0, 0, . . .x[n-1] = . . . 0, 0, 0, 1, 2, 3, 0, 0, . . .

(x[n] + x[n-1])/2 = . . . 0, 0, 1/2, 3/2, 5/2, 3/2, 0, 0, . . .

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© 2013

Quiz (2.3)

Compute the output of the system below when the input is

x [n] = δ[n] + 2δ[n − 1] + 3δ[n − 2]

x [n] + b y [n]

z−11/2

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© 2013

Answer

y [n] = 0 for n < 0

y [0] = x [0] + y [−1]/2 = 1

y [1] = x [1] + y [0]/2 = 2 + 0.5 = 2.5

y [2] = x [2] + y [1]/2 = 3 + 1.25 = 4.25

y [3] = x [3] + y [2]/2 = 0 + 2.175

. . .

y [n] = 4.25 · (0.5)n−2 for n ≥ 2

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© 2013

Quizzes

Q1 Write the recursive equation for the KS algorithm

Q2 A major reason for using the KS algorithm is that it is simple since it uses onemultiplication and addition per output sample. However, it requires M memory elements!Suggest an architecture for the violin synthesis which uses only M/2 memory elementsand write out an example for a small M, like M = 8.

Hint: Use M even, and think of up- and down-triangles!

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© 2013

Answers to questions

A1: yn = αyn−M+1 + xn

A2: Use the idea is that a single triangle can be used with alternating signs.

x [n] + b y [n]

z−M−1

For ex., for a period of 8 samples, M = 4, xn = 0, 1, 2, 1 and α = 1yn = 0, 1, 2, 1, 0,−1,−2,−1, 0, 1, 2, ....

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© 2013

digital signal processingquizzes q1 write the recursive equation for the ks algorithm q2 a major...

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