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Physics 401

Fourier Analysis

(week of Sep 27

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• On Units: deciBel db
• Fourier Series
• Discrete Fourier Transform
• Aliasing

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On Units: The deciBel

Alexander Graham Bell:

The human ear has logarithmic

response to power in a sound wave

Power

1m W

0.1mW

0.01mW

10m W

100m W

Hearing

0

- -

1

2

P (^) log P dB

20

10

0

reference

P

P

dB  10 log

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The Same Signal in Frequency Domain

1. 0005 sin( )

2. 001 sin( )

( ) sin( )

0 3 3

0 2 2

0 1 1

 

 

 

 

  

  

V t

V t

V t V t

Example:

Small signal components

are not visible in the time

domain!

t

ω

V(t)

[Volts]

V(

ω

)

[db]

ω 1

ω 2

ω 3

0 dB

-60 dB

-67 dB

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dB Scale for Voltages

10

2 V

10 V

1 V

10

3 V

10

4 V

V [Volts] (^) dB

80

60

40

20

( 1 Volt) 0

20 log

RMS

V

V

dB

10

- V

10

- V

10

- V - - -

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An Example (an odd function)

even k

vs odd k

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2

2

( ) ( )e

( ) ( )e

ift

ift

H f h t dt

h t H f df



• 

ò

ò

( ) ( )e

( ) ( )e

i t

i t

H h t dt

h t H d





ò

ò

frequency f in Hz (^) angular frequency  in rad / s

t

h(t)

ω

|H(ω)|

2

time domain frequency domain

Use Fourier Transform to Analyze

Continuous Functions

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If bandwidth in the signal is limited to frequencies below f c

i. e. H(f) = 0 for f > f c

then

sin 2

( )

c

n

n

f t n t

h t h

t n t t





é ù æ ö ê ú ç ÷ ê ú è ø ë û

æ ö

ç ÷ è ø

¥

-¥

• 



•  

å

Continuous function represented by an infinite series!

But in experiments we collect a finite number of samples!

The Sampling Theorem shows how to

construct the continuous function from discrete values

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k

h  h k  t k  N -

Discrete Fourier Transform

f

|H

n

2

1

0

exp( 2 / )

N

n k

n

h H ikn N

N



å

1

0

exp(2 / )

N

n k

k

H h  ikn N



å

 









 (^) 









 t

h k



  ^ 

 





 (^) 





max

T  N  t

c

f

t

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Aliasing Exercise: 1.263 kHz sine wave with 20 kSa/s and f c

= 10 kHz

Measured Frequency from FFT: 1.27 kHz

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Aliasing Exercise: 1.263 kHz sine wave with 10 kSa/s and f c

= 5.0 kHz

Measured Frequency from FFT: 1.27 kHz

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Aliasing Exercise: 1.263 kHz sine wave with 2.0 kSa/s and f c

= 1.0 kHz

NB aliasing

Measured Frequency from FFT: 0.736 kHz

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Aliasing Exercise: 1.263 kHz sine wave with 1.0 kSa/s and f c

= 0.5 kHz

NB aliasing

Measured Frequency from FFT: 0.263 kHz