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Signals

Outline

• Announcements:– Homework III: due Today by 5, by e-mail

• for P.4: n can be anything you want

– HW IV available soon.

• Binary Files• Signals, signals, signals

Binary Basics

• All computer files are “binary”, that is composed of 0’sand1’s

• When the computer reads ASCII files, it takes chunks of8 bits (1 byte) and looks up the character

• To save pi to 16 digits takes 18 bytes in ASCII• If you save the 1’s and 0’s that correspond to the

double precision value of pi, that takes only 8 bytes

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• You can’t just look at them• You must know exactly how they were

created– integers vs. floating point– single precision vs. double precision– signed vs. unsigned

Problem with Binary Files

• fid=fopen(fname,’r’);%’r’ = read binary• A=fread(fid,N,precision)

– N=number of data points, use Inf to readeverything

– precision is how the file was created• “uint64” is an unsiqned integer saved in 64 bits• “double” is a double

Reading Binary files

Free advice (you get whatyou pay for)

• The only reasons to use binary files are– someone gives you one– you enjoy frustration and pain– you’re too poor (or cheap) to buy a new

hard drive

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Writing .mat files outsideMatlab

• .mat files are a great format for storing data– easy to use with Matlab– multiple variables / file– compact

• It is possible to save data to .mat files fromC/C++ programs using Matlab C/C++ library

• For more info:– See Lecture 09 notes– Take CIS404!

Signals

• Signals are time series– Examples:

• Sound (pressure vs.time)

• Earthquake(displacement vs. time)

• S&P 500 ($/share vs.time)

• Signals are usuallycontinuous, but wesample them at discretetimes– regular vs. irregular

sampling– Sampling frequency

Signal Basics

• Simplest signal: s(t)=A*sin(2*pi/f*(t-phi))– A=amplitude– f=frequency– phi=phase

• A,f,phi summarize signal

1/f

A

phi

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Fourier Analysis

• Real signals are more complicated• Fourier proved that any function can be

represented as sum of sines & cosines ofvarious frequencies:

f=(k-1)/N

Fourier Analysis

s1(t)

s2(t)

0.5*s1(t)+s2(t)

f=8

f=1/2

Signals in MATLAB

• In MATLAB, a signal is a vector of numbers s• Matlab’s signal processing functions assume

– s was sampled regularly– s is complete (no missing data, nans, -999’s etc.)

• You must know sampling frequency f

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Fourier Analysis

• Fourier transform (fft)– Finds amplitudes over a range of frequencies

• amp=fft(s);

– If s is n-by-1, amp will be n-by-1 and complex•

– First half of amp contains info:• a=real([amp(1),2*amp(2:n/2)])/n; %cos coefs.• b=imag([0, -2*amp(2:n/2)])/n; %sin coefs.• f= (0:(n/2-1))/n/(t(2)-t(1)); %frequencies• F=2*pi*t(:)*f;• s2=cos(F)*a(:)+sin(F)*b(:); %original signal

Fourier Analysis

• What’s the point?– fft transforms from time-domain to

frequency domain• Energy at frequency j = sqrt(a(j).^2+b(j).^2)• Plot energy vs. f• Peaks are important f’s• Could remove energy at some frequencies

Signal Processing Toolbox

• Matlab’s Signal Processing Toolboxcontains lots of functions for workingwith digital signals– transforms beyond fft– filter design, implementation– spectral analysis– Check on-line help for more info– Need to understand theory better than I

do!