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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
-
Digital Signal Processing
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
sing
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
sing
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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1865 1890 1915 1940 1965 1990
year
m3/s
Representations of flood data: circa AD 2000
2.1 4
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Probably your first scientific experiment...
Daily temperature
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30
days
C
2.1 5
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Probably your first scientific experiment...
Daily temperature
0 500 1000 1500 2000 2500 3000
5
0
5
10
15
20
25
30
days
C
2.1 5
Digital Sig
nal Proces
sing
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
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
History and sociology
world population (billions)
1AD 1700AD 2030AD0
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5
6
7
8
9
year
2.1 7
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Economics
a purely man-made signal: the Dow Jones industrial average
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10000
1891 1916 1941 1966 1991 2016
year
2.1 8
Digital Sig
nal Proces
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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
b b b b b b b b b b b b b b b b
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1
2.1 10
Digital Sig
nal Proces
sing
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
sing
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 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 13
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
The Frankenstein switch...
2.1 14
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
The exponential decay
x [n] = |a|n u[n], |a| < 1
b b b b b b b b b b b b b b b b
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2.1 15
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
How fast does your coffee get cold...
2.1 16
Digital Sig
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
The sinusoid
x [n] = sin(0n + )
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2.1 18
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Oscillations are everywhere!
2.1 19
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Four signal classes
finite-length
infinite-length
periodic
finite-support
2.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Four signal classes
finite-length
infinite-length
periodic
finite-support
2.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Four signal classes
finite-length
infinite-length
periodic
finite-support
2.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Four signal classes
finite-length
infinite-length
periodic
finite-support
2.1 20
Digital Sig
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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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b
0 1 2 3 4 5 6 70
1
0 1 2 3 4 5 6 70
1
2.1 26
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: finite-support
x [n]
. . . x0 x1 x2 x3 x4 x5 x6 x7 . . .
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1 bb
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b
0 1 2 3 4 5 6 70
1
2.1 26
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: finite-support
x [n]
. . . 0 0 0 x0 x1 x2 x3 x4 x5 x6 x7 0 0 0 . . .
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1 bb
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b
0 1 2 3 4 5 6 70
1
2.1 26
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: finite-support
x [n 1]
. . . 0 0 0 0 x0 x1 x2 x3 x4 x5 x6 x7 0 0 . . .
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b
0 1 2 3 4 5 6 70
1
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0 1 2 3 4 5 6 70
1
2.1 26
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: finite-support
x [n 2]
. . . 0 0 0 0 0 x0 x1 x2 x3 x4 x5 x6 x7 0 . . .
b
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b
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b
0 1 2 3 4 5 6 70
1
b b
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0 1 2 3 4 5 6 70
1
2.1 26
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: finite-support
x [n 3]
. . . 0 0 0 0 0 0 x0 x1 x2 x3 x4 x5 x6 x7 . . .
b
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b
0 1 2 3 4 5 6 70
1
b b b
b
b
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b
0 1 2 3 4 5 6 70
1
2.1 26
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: finite-support
x [n 4]
. . . 0 0 0 0 0 0 0 x0 x1 x2 x3 x4 x5 x6 . . .
b
b
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b
b
b
b
b
0 1 2 3 4 5 6 70
1
b b b b
b
b
b
b
0 1 2 3 4 5 6 70
1
2.1 26
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: periodic extension
[x0 x1 x2 x3 x4 x5 x6 x7]
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
0 1 2 3 4 5 6 70
1
2.1 27
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: periodic extension
x [n]
. . . x0 x1 x2 x3 x4 x5 x6 x7 . . .
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1 bb
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
2.1 27
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: periodic extension
x [n]
. . . x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 x7 x0 x1 x2 . . .
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1 bb
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
2.1 27
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: periodic extension
x [n 1]
. . . x4 x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 x7 x0 x1 . . .
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
b
b
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b
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b
0 1 2 3 4 5 6 70
1
2.1 27
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: periodic extension
x [n 2]
. . . x3 x4 x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 x7 x0 . . .
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
2.1 27
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: periodic extension
x [n 3]
. . . x2 x3 x4 x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 x7 . . .
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
2.1 27
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
Shift of a finite-length: periodic extension
x [n 4]
. . . x1 x2 x3 x4 x5 x6 x7 x0 x1 x2 x3 x4 x5 x6 . . .
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
b
b
b
b
b
b
b
b
0 1 2 3 4 5 6 70
1
2.1 27
Digital Sig
nal Proces
sing
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
sing
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
sing
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
nal Proces
sing
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
nal Proces
sing
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
-
Digital Signal Processing
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
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
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
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
The complex exponential
e j = cos+ j sin
1 1
1
1
Re
Im
ej
b
2.2 37
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
The complex exponential
z: point on the complex plane
Re
Im
zb
2.2 38
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
The complex exponential
rotation: z = z e j
Re
Im
zb
zb
zb
2.2 38
Digital Sig
nal Proces
sing
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
sing
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
b
x[1]b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
x[2]b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
x[3]b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
x[4]b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
x[5]b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
b
x[6] b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
b
b
x[7]b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
b
b
b
x[8]
b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
b
b
b
b
x[9]
b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
b
b
b
b
b
x[10]
b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
b
b
b
b
b
b
x[11]b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
b
b
b
b
b
b
b
x[12]b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
b
b
b
b
b
b
b
b
x[13]b
2.2 39
Digital Sig
nal Proces
sing
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
b
b
b
b
b
b
b
b
b
b
b
b
b
b
x[14]b
2.2 39
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
x[0]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
x[1]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
x[2]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
x[3]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
x[4]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
x[5]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
x[6] 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
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
Careful: not every sinusoid is periodic in discrete time
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
-
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
-
One point, many names
Re
Im
ej
b
2pi +
2.2 43
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
One point, many names
Re
Im
ej
b
6pi +
2.2 43
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
One point, many names
Re
Im
ej
b
2.2 44
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
-
One point, many names
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
-
Lets go really too fast
= 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
-
Lets go really too fast
= 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
-
Lets go really too fast
= 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
-
Lets go really too fast
= 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
-
Lets go really too fast
= 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
-
Lets go really too fast
= 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
-
Lets go really too fast
= 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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
Digital Signal Processing
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
-
Building Blocks: Adder
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
-
Building Blocks: Adder
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
-
Building Blocks: Multiplier
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
-
Building Blocks: Multiplier
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
-
Building Blocks: Unit Delay
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
-
Building Blocks: Unit Delay
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
-
Building Blocks: Arbitrary Delay
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
-
Building Blocks: Arbitrary Delay
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
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
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] = u[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 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
-
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
-
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
-
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
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
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
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
b b b b b b b b b 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
b b b b b b b b b 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
b b b b b b b b b 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
b b b b b b b b b 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
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
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
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
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
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
b b b b b b b b b 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
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
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
b
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b
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
b
b
b
b
b
b
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b b b
b
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b
b
b
b
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
-
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
-
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
b
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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
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b
b
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
b
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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
b
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