Digital Signal Processing A Merger of Mathematics and Machines

Digital Signal ProcessingDigital Signal ProcessingA Merger of Mathematics and A Merger of Mathematics and

MachinesMachines

2002 Summer Youth ProgramElectrical and Computer Engineering

Michigan Technological University

Department of Electrical and Computer Engineering

Copyright Michigan Technological University

2002 Summer Youth Program

Signals and Sounds

The simplest signal is the sinusoid:

t

t

t

t

frequency = 500 Hz

frequency = 1 KHz

frequency = 2 KHz

frequency = 4 KHz




Signals and Sounds

Sums of sinusoids

0 0.005 0.01 0.015 0.02-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (seconds)

ampl

itude

))(cos())(cos()( t12092t9412tx

941 1209 f

‘spectral’ representation




Signals and Sounds

1 2 3

4 5 6

7 8 9

* 0 #

697 Hz

941 Hz

852 Hz

770 Hz

1209

Hz

1336

Hz

1477

Hz

s t f t f trow column( ) cos( ) cos( ) 2 2

Dual-tone multiple frequency (DTMF)

Frere Jacques

Olympic Fanfare




Signals and Sounds

time (seconds)

fre

que

ncy

(Hz)

2 4 6 8 10 12

0

500

1000

1500

2000

2500

3000

3500

4000

Olympic Fanfare




Signals and Sounds

What other signals (or sounds) can we make from sinusoids?

– Answer: ALL OF THEM!

This is Fourier theory and it forms the basis for many branches of electrical engineering.




The Common Loon




Pied-Billed Grebe




Tundra Swan (Whistling Swan)




Signals and Sounds

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (seconds)

ampl

itude

250 1250

750 1750

2250 f

Signal Spectrum




Signals and Sounds

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (seconds)

ampl

itude

250 f

Signal Spectrum

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (seconds)

ampl

itude




Filtering

One of the key concepts in signal processing is the idea that systems can be built to analyze or modify a signal’s spectrum.

– Applications:• speech recognition• speaker recognition• noise removal




Filtering

+

noise




Filtering

0 500 1000 1500 2000 2500 3000 3500 4000 45000

50

100

150

200

250

300

frequency (Hz)

gain

frequency response

H(f)




Filtering

time (seconds)

fre

que

ncy

(Hz)

0.5 1 1.5 2 2.5 3 3.5 4

0

2000

4000

6000

8000

10000

+

noise

time (seconds)

fre

que

ncy

(Hz)

0.5 1 1.5 2 2.5 3 3.5 4

0

2000

4000

6000

8000

10000




Filtering

H(f)time (seconds)

fre

que

ncy

(Hz)

0.5 1 1.5 2 2.5 3 3.5 4

0

2000

4000

6000

8000

10000

time (seconds)

fre

que

ncy

(Hz)

0.5 1 1.5 2 2.5 3 3.5 4

0

2000

4000

6000

8000

10000

0 2000 4000 6000 8000 10000 120000

0.5

1

1.5

2

2.5

3

3.5

frequency (Hz)

gain

frequency response




Digital Signal Processing

A-to-D D-to-ADSPanalog analogdigital digital

Analog signals are continuous in time and amplitude.

Digital signals are discrete in time and amplitude.




Sampling

16 bits gives 216 = 65,536 amplitude levels

8 bits gives 28 = 256 amplitude levels

4 bits gives 24 = 16 amplitude levels




Sampling of Sound

16 bits (CD quality) 12 bits 8 bits (phone quality) 16 bits / 8 bits 8 bits / 6 bits 8 bits / 4 bits 8 bits / 2 bits




Digital Signal Processing

Digital filters are really simple!– four-sample moving average filter

– recursive (feedback) filter

44nx2nx1nxnx

ny)()()()(

)(

)()(.)( nx1ny90ny




Pictures Too!




So What Do You Need To Learn?

Signal and System Theory– Spectral analysis– Filter design

Digital Signal Processing– Software systems– Hardware systems

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Digital Signal Processing A Merger of Mathematics and Machines

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