Objectives of Lecture on DSP

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"Objectives" of Lecture on DSP

• Introduction to laboratory #1• The need for DSP• Resolution, Aliasing & Windowing• Introduction to FFT.

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Why Digital Signal Processing?

• Allows fast & complex calculations, usuallywithout loss of information.

E.G.– MRIs

– Communication (TV,CD,MP3, etc.)– Signal diagnostics (machine health, “looking” for

hidden details)

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A/D converters(Analogue/Digital)

This lecture, 3 concepts & 1 process.

•A/D resolution

•Aliasing (sampling rate)

•Windowing (leakage)

Time domain

Frequency domain

FFT

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Getting it into the computer!(A/D)

x(t)

time, t

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Getting it into the computer!(A/D)

x(t)

time, t

We now havea sequence ofnumbers, i.e., avector.

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Some definitions

Δt

•Sampling rate

fs=1/Δt

•Window length orTime Record length,T.R.

T.R.

x(t)

t

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Mention of one other issue:Resolution

For a specific voltage range there will onlybe a certain number of "divisions" available (see later).

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Mention of one other issue:Resolution

Δvoltage

For a specific voltage range there will onlybe a certain number of "divisions" available (see next slide).

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Example of a A/D spec.

“12bit A/D converter with a sampling rate of 15kHzover a range of -/+ 5 volts”

Voltage increments:Δvoltage = 10/212 = 10/4096 = 2.441mVTime increment:Δt = 1/15,000 sec.

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Aliasing

• How fast do we have to sample?• What happens if we don't sample fast

enough?• What can we do about it?

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Fast enough?

1Hz analogue signal10Hz sampling frequency

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Or is this fast enough?

1Hz analogue signal5 Hz sampling frequency

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1Hz analogue signal or a 4Hz analogue signal5 Hz sampling frequency

Aliased signal

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We must sample AT LEAST twice as fast as the highestfrequency that is present.e.g. here, 4Hz signal therefore sample 8Hz.

!

>

“Onset” of aliasing

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• To avoid aliasing, the analogue signal isfirst FILTERED (anti-aliasing filters) toensure that all frequencies higher than 1/2the sampling frequency have been removed.

• The maximum detectable frequency issometimes called the Nyquist frequency.

SUMMARY

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WindowingThere must be a finite record length.

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Windowing

T.R.

Finite record length,hence “stop” and “start”.

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Windowing

The Fourier series algorithim believes thisfinite record, of length T.R., to be thefundamental part of a periodic function. Itattempts to recreate a function as sketchedbelow. The sudden “stop” and “start”,shown in red, causes the creation ofadditional components in the frequencydomain. These additional frequencies areknown as LEAKAGE. This can beminimized by tapering the beginning andending of each T.R. to zero.

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*

Windowing function

Time record

Windowed time record

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Quick introduction toFast Fourier Transform (FFT)

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Recap Fourier series.

x t( ) =1

Ta0

+ ancos !

nt( ) + b

nsin !

nt( )

n=1

"

#$ % &

' &

( ) &

* &

Periodic signal of period τ, x(t) = x(t + !)

where !n=2n"

#n = 1,2, 3,.....

an=2

!x(t)cos("

nt)dt

#! / 2

! / 2

$

bn=2

!x(t)sin("

nt)dt

#! / 2

! / 2

$

and where

See eq(1.2.1-3)

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Fourier series and FFT

x(t) =4

!sin("1t) +

4

3!sin(3"1t) +

4

5!sin(5"1t) + ....

Recall that for the following square wave:

x(t)

t

1

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Another way of viewing this:x(t)

t

frequency

4

!

4

3!4

5!

!12!

13!

1

Magnitude of Sine component

Time domain

Frequency domain

FFT

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Lab. Example

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FFT summary

T.R.

Δt

time

frequency

Mag.1/T.R.

0

1/2Δt

N data points N/2 Mag. data pointsN/2 Phase. data points

** *

**

*

FFT