Q&A (Labs 4&5)

School of Architecture, Civil and

Environmental Engineering

EPFL, SS 2009-2010

http://disal.epfl.ch/teaching/signals_instruments_systems/

Outline

• Lab 4 Concepts:

– Discrete signals, signal processing in Matlab

– Convolution

– Fourier series

• Lab 5 Concepts:

– High- / Low- pass filter

– Bode diagrams

Discrete Convolution

m

m

mgmnftgf )()()(

Indeces:

n: shift of reflected signal

m: sum over m of multiplied values

Example: Moving Average

Ex.: Moving Average II

L = 5 L = 20 L = 60

L

1/Lsignal ‘rect’:

This moving average is a convolution of rectangular signal (of length L) with

a noisy sinus signal. Does this look like a filter? If yes, which one?

Moving Average in Frequency and

Time-Domain

signalrect

)()( signalfftrectfft

time-domain

frequency-domain

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-10 -8 -6 -4 0 2 4 6 8 10

0.2

0.4

0.6

0.8

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

The result is equivalent!

Using Matlab

% moving average example

clear; figure; hold;

xs = 0:100;

ys = 1*sin(xs*0.1) + 0.5*sin(xs) +

0.9*sin(xs*2.3);

plot(ys,'k','LineWidth',2);

xf = 0:10;

l = 60;

yf = repmat((1/l),1,l);

yc = conv(yf,ys);

plot(yc,'r','LineWidth',2);

legend('Original signal','Convolution');

Using Matlab

% moving average example

clear; figure; hold;

xs = 0:100;

ys = 1*sin(xs*0.1) + 0.5*sin(xs) +

0.9*sin(xs*2.3);

plot(ys,'k','LineWidth',2);

xf = 0:10;

l = 60;

yf = repmat((1/l),1,l);

yf = [repmat(0,1,20), yf];

yc = conv(yf,ys);

plot(yc,'r','LineWidth',2);

legend('Original signal','Convolution');

The rectangular signal is shifted (the origin is at position i=21).

The origin of the convolution is (i+j-1)=(21+0-1)=20.

Lab 5 – Q2, Q3

Original signal and its FFT

In Matlab, the resulting

FFT amplitude is A*n/2,

where A is the original

amplitude and n is the

number of FFT points.

Lab 5 – Q2, Q3

2Hz sampling

Lab 5 – Q2, Q3

4Hz sampling

Lab 5 – Q2, Q3

20Hz sampling

What does the Nyquist-Shannon

theorem say?

Lab 5 – Q4

Original signal

Lab 5 – Q4

What is this effect called?

Can you explain the FFT peak at 3Hz?

Lab 5 – Q5

Original signal

Lab 5 – Q5

10Hz sampling, low-pass with 3Hz cutoff

Lab 5 – Q5

Bode plot

Lab 5 – Q7

10Hz sampling, 1st order BW low-pass with 3Hz cutoff

Lab 5 – Q7

10Hz sampling, 3rd order BW low-pass with 3Hz cutoff

Lab 5 – Q7

~1*20dB ~3*20dB

1st order 3rd order

attenuation per decade ~ n*20dB

Lab 5 – Q8

Butterworth of order n: attenuation per decade ~ n*20dB

Lab 5 – Q12,Q13

freq

uen

cy

frequency

amplitude

What is the FT of a rectangular function? And in 2D?