fourier lab

8/3/2019 Fourier Lab

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Signals and Systems

Laboratory Exercise 5

Fourier Spectra

Objective

The objective of this lab is to illustrate and understand the fundamental concepts of Fourier

spectra, investigate their properties, and practical applications.

The prelab should be done before the lab, and the answer sheet, not longer than one page,

should be given to the demonstrator to sign before the lab starts. You will not be allowed to do

the prelab during the Lab.

Theoretical background is provided in the textbook references (check the website) and in the

lecture notes. Fundamental formulas are mentioned in the introductory part of this handout.

Your lab report should have a title, a short description of the purpose of the lab, and include

the prelab report. For the experiments, include sketches or plots of the models (be sure that youlabel them and refer to them appropriately if necessary), relevant details of the experiment, and

plots. The text part of the report, with answers, comments and conclusions, should not be more

than 1 page long.

Introduction

1 Fourier Series

A continuous-time periodic signal with periodTcan be represented as an infinite sum of complex

exponentials, multiplied by appropriate Fourier coefficients:

x(t) =+

Xkejk 2

Tt (1)

where the coefficients Xk are calculated as:

Xk =1

T

+T/2

T/2

x(t)ejk2

Ttdt (2)

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Coefficient X0 is the mean value of the signal on one period (or its DC component). Coeffi-cients with other indices represent higher harmonics of the signal. The signals spectrum can be

viewed as a series of impulse functions in continuous frequency.

2 Fourier Transform

Fourier transform is used to represent spectra for signals that are continuous in time and aperiodic

(which is the case for most practical signals). The Fourier transform of signal x is calculated as:

X() =

+

x(t)ejtdt (3)

where = 2f is the angular frequency. The signal itself can be calculated from its fre-quency domain representation as:

x(t) =1

2

+

X()ejtd (4)

These relations can be seen as a limit case of Fourier series, when the period is allowed to ap-

proach infinity, leading to a continuum of frequency components (1/T, which is the fundamentalfrequency converges to zero). The spectrum is a continuous function of frequency.

3 Discrete-Time Fourier Transform

When dealing with an aperiodic signal in discrete time, the discrete-time Fourier transform

(DTFT) is used to represent the spectrum. This transform is expressed as:

X(ej) =+

x[n]ejn (5)

Brackets [ ] were used here to indicate a function that is discrete in time. = 2fFs

is the

normalised angular frequency (Fs is the sampling frequency of the discrete-time signal). As youknow from sampling theory, this spectrum is periodic in . It is also a continuous function offrequency. The signal can be reconstructed from its frequency domain representation as:

x[n] =1

2

+

X(ej)ejnd (6)

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4 Discrete Fourier Transform

Discrete Fourier transform is used when the signal is discrete-time and periodic. In practice, it is

used to calculate frequency domain representation of aperiodic signals in a given time interval,

by assuming their periodic extension. DFT is calculated as:

X[k] =N1n=0

x[n]ej2k

Nn (7)

The signal is reconstructed by using the inverse discrete Fourier transform (IDTFT):

x[n] =1

N

N1k=0

X[k]ej2n

Nk (8)

The definition of DFT can also be written in matrix form X= WN x, where:

WN =

1 1 1 1

1 11N 12N

1(N1)N

1 21N 22N

2(N1)N

......

......

1 (N1)1N

(N1)2N

(N1)(N1)N

, (N = e

j 2N ) (9)

and x is a column-vector containing the signal values. The equation for signal reconstructioncan analogously be written in matrix form.

Since the signal is periodic in time, the spectrum is discrete in frequency, and because thesignal is discrete in time the spectrum is also periodic in frequency. So in this case both the

signal and the spectrum are discrete and periodic in their own domains. As the DFT of a signal

is periodic, with period 2, the spectrum is calculated only on the interval [0, 2]. This intervalcorresponds to the interval [0, Fs] in the physical frequency domain. The fact that the spectrumin this case only exists in discrete frequency points, means that frequency components of a signal

that do not fall on those discrete frequency values cannot be accurately resolved. Their power will

be distributed between adjacent frequency points. Assuming that the number of signal samples

in one period is same as the number of frequency samples in DFT spectrum, the frequency

resolution of a DFT spectrum is given as f = Fs/N. To improve resolution, we therefore need

to increase N or reduce Fs. As mentioned above, DFT is often used to represent the spectrum ofaperiodic signals, by picking a finite number of samples N, and assuming that those samples were

taken from a periodic signal. This process is equivalent to multiplying the signal with a function

that is non-zero only in a finite number of points (window function). This process of windowing

and has a side-effect of transferring some of the components power to other frequencies known

as spectral leakage, which reduces the ability to resolve individual frequency components, as well

as the ability to accurately calculate their amplitudes. For instance the rectangular window, if not

applied carefully, will introduce discontinuities in the waveform which will result in bad results

in DFT. By choosing different window functions (like Hamming, Hann, Flattop etc.), things can

be improved, but a trade-off between resolution and amplitude accuracy is always present.

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Prelab

A

x(t)

tT

t

Figure 1: Ideal low-pass signal spectrum

PQ.1 For the pulse train signal shown in Figure 1, complex Fourier coefficients (for k = 0) canbe calculated as:

Xk = Asin(k)

k

where = T

is the duty cycle. For k = 0, X0 = A. Write Matlab code to calculatecoefficients for the following cases (Assume A = 1):

a) first 60 coefficients with positive indices when T = 0.1 and = 0.05

b) first 60 coefficients with positive indices when T = 0.1 and = 0.005

c) first 300 coefficients with positive indices when T = 0.5 and = 0.005

d) first 300 coefficients with positive indices when T = 0.5 and = 0.001

Plot coefficients for all these cases against frequency in one figure by using the subplot

and stem commands. Extra marks will be awarded if you dont use the for loop to

calculate coefficients themselves.

PQ.2 For the spectrum shown in Figure 2, calculate its inverse Fourier transform. (Hint: use

manipulations similar to those used in the reconstruction tutorial). Sketch the time domain

signal.

1

wW-W

X( )w

Figure 2: Ideal low-pass signal spectrum

PQ.3 In Matlab, create a 1 second time vector with N elements, where N = 64 (hint: check the

linspace function), and a sinewave of frequency 25.2 Hz. Plot this signal against time.

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Use fft command to calculate thr N point DFT of the signal you created. Measure how

long it takes (use tic and toc commands).

Create a WN vector as described above. It can be done in just one line of code, althoughyou dont have to do it that way. (hint: Create a matrix of exponents use the colon op-

erator and the fact that a n 1 column vector multiplied by a 1 n row vector producesa n n matrix; then use element-wise exponential function). Calculate the DFT of yoursignal by multiplying its transpose with WN matrix. (In fact the whole operation of cal-culating DFT can be done in one line of code). Check how long it takes to generate WNmatrix and multiply it with the signal. Compare the speed with Matlabs fft routine. Note

both times.

Plot magnitudes of both calculated DFTs in one figure using the subplot command. esti-

mate how different the two DFTs are by using the norm command on their difference.

Note that you will need to transpose the result of the first calculation to do this (use .

operator for ordinary transpose).

Plot both calculated spectra (for power spectral densities you will need to plot the squared

magnitude of the DFT) in one figure by using the subplot command.

Repeat the calculations for N = 128, 256, 512, 1024, note how the calculation times

change. Plot the ratio of calculation times vs. the number of DFT points. What does

it tell you?

1 Lab Work

LQ.1 Set the output of the function generator to high Z because the input impedance of the

scope is 1 M. Generate a sinewave with frequency 1 kHz and amplitude 2 Vpp. PresAutoset on the scope, and verify the parameters of the signal. Set the timebase to 200

s/div. In the math menu choose fft, and set windowing function to hanning. For thisand all following tasks set readout to abs. Comment on what you see. Use the cursor

to verify the frequency and the magnitude.

LQ.2 Change the timebase to 5 ms/div and observe what happened. How did this affect the

resolution? Why? (hint: the scope calculates a 512-point FFT of exactly what is on the

screen, so we have a larger number of periods i.e. a larger time interval with the samenumber of samples).

LQ.3 Change the windowing function to rectangle and set the timebase so that you see

only one period of the signal on the screen. Observe the resulting FFT. Now change the

sinewave frequency to 1.1 kHz so that a little more than one period is shown on the screen.

How did the FFT change? Remember that the FFT is calculated for the periodic extension

of the captured signal. In this case, is the FFT calculated for a sinusoidal wave?

LQ.4 Set the modulation to FSK, with FSK rate of 10 Hz and FSK hop of 100 Hz. Set the

timebase to 5 ms/div. Press Run/Stop to capture one screen of data. How many fre-

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quencies are present in the captured signal? Determine the frequencies of the sinusoids

from the time-domain display. Switch to FFT display and measure the frequencies of the

components. Compare.

LQ.5 Set the output to a square wave with frequency 1 kHz and amplitude 1 Vpp with a +0.5

V DC offset. Set the timebase on the scope to 500 s/div. Show the FFT of this signal onscreen (you can turn of the time-domain display). Use the cursor to identify the frequencies

of harmonics. Describe the spectrum. Can you compare it to any of the results from your

prelab?

LQ.6 Switch to a pulse output, and change the duty cycle from 50% to 10%. Observe the result.

Was it expected? How does this relate to the prelab results? Change the timebase from

500 s/div to 200 s/div. What does the result remind you of?

LQ.7 Change the generators output to a Sin(x)/x function, keeping the frequency at 1 kHz, and

change the timebase to 200 s/div. Observe the FFT. Is this result expected? Why?

Report

Prelab

PQ.1 Submit your Matlab code. Attach the plot.

PQ.2 Submit your calculation and hand sketch of the signal.

PQ.3 Submit your Matlab code. Submit the plot of both power spectral densities for N=64.

Submit the plot of the ratio of the calculation times. Comment on it.

Lab Exercise

In no more than one page, answer the lab questions.

75% of the grade will be awarded for correct answers, and up to 25% will be awarded for the

quality of answers.

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