DSP for Dummies aka

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How to turn this (actual raw sonar trace)

Into this .. (filtered sonar data)

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Fundamental property of all analog signals

complex signals

i.e. audio

i.e. digital bitstream

Decompose

into summation

of sinusoids

amplitude

phase

frequency

properties

Fourier Transform

How do we analyze the frequency components of a complex signal

Time space x(t)

Frequency space X()

single frequency signal 0

t

0

t

X() is complex -- complex conjugate encodes phase

Fourier transform is invertable

Some properties

Digital Signals

Sample 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 Transform

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 noise

What 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 inifinity

to 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

xt = Cii=0

i=(taps1)

*Sti

FIR filterA special type of weighted average designed to enhance signal

components at some frequencies while attenuating others

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+ Xt

Selecting the coefficients tunes

the filter to pass or attenuated specific

frequency bands

Selecting the coefficients a.k.a designing the filter

Matlab demo

How a filter works (theoretically)

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signal

noise

FFT

noise

minus

FFT-1

signal

Sample Code

/*

fir.h

simple fir filter code

* dmc - 03-04-08

* for cs 1567 note this code does not set up coefficents

*/

#define TAPS 4 // how many filter taps

typedef struct

{

float coefficients[TAPS];

unsigned next_sample;

float samples[TAPS];

} filter;

/* firFilterCreate()

* creates, allocates, and iniitializes a new firFilter

*/

filter *firFilterCreate()

{

int i;

filter *f = malloc(sizeof(filter));

for (i=0; isamples[i] = 0;

f->coefficients[i] = 1. /(float) TAPS; // user must set coef's

}

f->next_sample = 0;

}

/* firFilter

* inputs take a filter (f) and the next sample (val)

* returns the next filtered sample

* incorporates new sample into filter data array

*/

float firFilter(filter *f, float val)

{

float sum =0;

int i,j;

/* assign new value to "next" slot */

f->samples[f->next_sample] = val;

/* calculate a weighted sum

i tracks the next coeficeint

j tracks the samples w/wrap-around */

for( i=0,j=f->next_sample; icoefficients[i]*f->samples[j++];

if(j==TAPS) j=0;

}

if(++(f->next_sample) == TAPS) f->next_sample = 0;

return(sum);

}

Filtered Wheel Encoder Data

Left Wheel

Right Wheel

Filtered NorthStar data

X Position

Y Position

Theta