Digital Signal Processing

Module 1: IntroductionDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

What is a signal?

Description of the evolution of a physical phenomenon

Examples:

temperature (weather)

pressure (sound)

magnetic deviation (recorded sound)

gray level on paper (photograph)

...

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Paolo Pran

doni and M

artin Vett

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2013

Digital Sig

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Paolo Pran

doni and M

artin Vett

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uu

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

Key ingredients:

discrete time

discrete amplitude

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Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

The circle point out (from left to right): Pythagoras, Parmenides, Plato,

Euclid. Green for philosophers, red for mathematicians

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

Digital Sig

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Paolo Pran

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artin Vett

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A B

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A BC

Digital Sig

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Paolo Pran

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artin Vett

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A BC D E FG

Digital Sig

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Paolo Pran

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artin Vett

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2013

A BC D E FG

n=1

1

2n= 1

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Idea here is: you got a point, nameless geometrical entity. Descartes comes and

puts a reference around it, point gets a name. Then more complex things get

names (such as lines) and can be described in terms of algebra

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

01

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0 1 2 3 4 5 6 7 8 9 10x

y

(6, 4)

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

01

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0 1 2 3 4 5 6 7 8 9 10x

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y = x + c

Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Calculus: ideas at the service of war

b

b

b

b

b b

b

b

b

b

~x(t) = ~v0t + (1/2)~g t2

Galileo, 1638

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Calculus: ideas at the service of war

b

b

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b b

b

b

b

b

~x(t) = ~v0t + (1/2)~g t2

Galileo, 1638

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Continuous-time signal processing: the average

a b

x =1

b a

b

a

f (t)dt

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Continuous-time signal processing: the average

a b

x =1

b a

b

a

f (t)dt

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Discrete-time signal processing: the average

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0 N 1

x =1

N

N1n=0

x [n]

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Discrete-time signal processing: the average

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0 N 1

x =1

N

N1n=0

x [n]

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Are we missing pieces?

What if the signal is too fast?

f (t) =?

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Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

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2013

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

The Sampling Theorem (1920)

Under appropriate slowness conditions for x(t) we have:

x(t) =

n=

x [n]sin((t nTs)/Ts)

(t nTs)/Ts

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

The world is analog, the computer is digital

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Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

So, what is resolution, really?

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

Key ingredients:

discrete time

discrete amplitude

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

x(t)

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

x(t)

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital Signals

b b

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x [n]

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital amplitude

Why it is important:

storage

processing

transmission

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Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

Data storage

Analog storage:paper, wax cylinders, reel-to-reel, vinyl, compact cassette, VHS, Betamax, silver plates,

Kodachrome, Super8, 8-Track, microfilm, ...

Digital storage:

{0, 1}

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Data storage

Analog storage:paper, wax cylinders, reel-to-reel, vinyl, compact cassette, VHS, Betamax, silver plates,

Kodachrome, Super8, 8-Track, microfilm, ...

Digital storage:

{0, 1}

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Data storage

Analog storage:paper, wax cylinders, reel-to-reel, vinyl, compact cassette, VHS, Betamax, silver plates,

Kodachrome, Super8, 8-Track, microfilm, ...

Digital storage:

{0, 1}

1 23

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Data storage

Analog storage:paper, wax cylinders, reel-to-reel, vinyl, compact cassette, VHS, Betamax, silver plates,

Kodachrome, Super8, 8-Track, microfilm, ...

Digital storage:

{0, 1}

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

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2013

Processing

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Paolo Pran

doni and M

artin Vett

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2013

Data transmission

TX channel RX

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

What happens to analog signals

x(t) + x(t)

(t)

1/G

x(t) = x(t)/G + (t)

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

What happens to analog signals

5

0

5

x(t)

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

What happens to analog signals

5

0

5

x(t)/G

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

What happens to analog signals

5

0

5

x(t)/G + (t)

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

We can amplify to compensate attenuation

x(t) + x1(t)

(t)

1/G G

but: x1(t) = x(t) + G(t)

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x(t)

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x(t)/G

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x(t)/G + (t)

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x1(t) = G [x(t)/G + (t)] = x(t) + G(t)

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Digital Sig

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Paolo Pran

doni and M

artin Vett

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2013

Transmitting a signal overseas

For a long, long channel we need repeaters

x(t) + x1(t) + . . . + xN(t)

(t) (t) (t)

1/G G 1/G G 1/G G

xN(t) = x(t) + NG(t)

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x(t)

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x(t)/G

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x(t)/G + (t)

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x1(t) = G [x(t)/G + (t)] = x(t) + G(t)

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x1(t)/G + (t)

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

x2(t) = G [x1(t)/G + (t)] = x(t) + 2G(t)

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of analog signals

5

0

5

xN(t) = x(t) + NG(t)

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

In digital signals we can threshold

x(t) + | | x1(t)

(t)

1/G G

x1(t) = sgn[x(t) + G(t)]

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of quantized signals

5

0

5

x(t)

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of quantized signals

5

0

5

x(t)/G + (t)

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of quantized signals

5

0

5

G [x(t)/G + (t)] = x(t) + G(t)

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Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013

Transmission of quantized signals

5

0

5

x1(t) = G sgn[x(t) + G(t)]

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Digital Sig

nal Proces

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital data throughputs

Transatlantic cable:

1866: 8 words per minute (5 bps)

1956: AT&T, coax, 48 voice channels (3Mbps)

2005: Alcatel Tera10, fiber, 8.4 Tbps (8.4 1012 bps)

2012: fiber, 60 Tbps

Voiceband modems

1950s: Bell 202, 1200 bps

1990s: V90, 56Kbps

2008: ADSL2+, 24Mbps

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Digital Sig

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Paolo Pran

doni and M

artin Vett

erli

2013

Digital data throughputs

Transatlantic cable:

1866: 8 words per minute (5 bps)

1956: AT&T, coax, 48 voice channels (3Mbps)

2005: Alcatel Tera10, fiber, 8.4 Tbps (8.4 1012 bps)

2012: fiber, 60 Tbps

Voiceband modems

1950s: Bell 202, 1200 bps

1990s: V90, 56Kbps

2008: ADSL2+, 24Mbps

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Paolo Pran

doni and M

artin Vett

erli

2013

Final words

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Paolo Pran

doni and M

artin Vett

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2013

END OF MODULE 1

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

2013