DSP for Dummiesaka

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How to turn this (actual raw sonar trace)

Into this .. (filtered sonar data)

Fundamental property of all analog signals

complex signals

i.e. audio

i.e. digital bitstream

Decomposeinto summationof sinusoids

amplitude

phase

frequency

properties

Fourier TransformHow do we analyze the frequency components of a complex signal

Time space x(t) Frequency space X()

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

single frequency signal

t

t

• X() is complex -- complex conjugate encodes phase• Fourier transform is invertable

Some properties

Digital SignalsSample amplitude at discrete time intervals

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Nyquist limit (http://www.medcyclopaedia.com)

(Harry Nyquist, 18891976, Swedish - American physicist), the maximum frequency of a signal that can be measured with a method that employs sampling of the signal with a specific frequency, the sampling frequency. According to Shannons sampling theorem, a signal must be sampled with a frequency at least twice the frequency of the signal itself. The maximum measurable frequency the Nyquist limit or frequency is thus half the sampling frequency. If the signal frequency is higher than the Nyquist limit, aliasing occurs.

Discrete Fourier TransformQuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.

Given a signal represented as a time sequence of samples, the DFT gives us a seqence of

frequency/phase amplitudes

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Signals and noiseWhat is noise? Any signal other than the one you are interested in!

Sources of Noise (the usual suspects)• statistical signals from active electronic components• crosstalk from other channels or other signals in the same channel• signals sensed from external sources (power supply, EM radiation)

Trival Example

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Signal of interest +

Noise signal =

Noisey Signal

Signal to Noise ratio

The relative amplitude of the signal of interest o the noise signal

Filtering out the Noise

Finite Impulse response (FIR) filter

Ideal Pulse (time domain)

t

Ideal Pulse (frequency domain)

to inifinityto inifinity

Ideal Pulse (time domain)

Zero rise/fall time

actual rise/fall time finite band of component frequencies

The number and values of the component freqencies is related to the rise/fall time of the pulse

http://www.chem.uoa.gr/Applets/AppletFourAnal/Appl_FourAnal2.html

How a filter works

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signal

noise

FFT

noise

minus

FFT-1

signal