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LABORATORY WORK №3
FOURIER TRANSFORM. SPECTRAL ANALYSIS
BRIEF THEORETICAL REVIEW
FOURIER ANALYSIS
The theory of Fourier series lies in the idea that most signals, and all engineering signals, can be represented as a sum of sine waves (including square waves and triangle waves). This analysis can be expressed as a Fourier series. There are two types of Fourier expansions:
FOURIER TRANSFORM
The Fourier transform is very similar to the Laplace transform. The Fourier transform uses assumption that any finite time-domain can be broken into a finite sum of sinusoidal (sine and cosine). Under this assumption, the Fourier transform converts a time domain signal into its frequency domain representation as a function of the radial frequency. The frequency domain representation is also called the spectrum of the signal.
The Fourier transform of a signal is defined as follows:
(1)
It is possible to show that the Fourier transform is equivalent to the Laplace transform when the following condition is true:
then
(2)
The formulas (1) and (2) represent direct Fourier transform.
The inverse Fourier transform is defined as follows:
(3)
The Fourier transform exists for all functions that satisfy the following condition
.
For arbitrary signals, the signal must be digitized, and a Discrete Fourier transform (DFT) performed. The standard numerical algorithm used for the DFT is called Fast Fourier Transform (FFT) or Discrete FFT (DFFT). The fast Fourier transform is a mathematical method for transforming a function of time into a function of frequency. Sometimes it is described as transforming from the time domain to the frequency domain. It is very useful for analysis of time-dependent phenomena.
FOURIER SERIES REPRESENTATION OF A PERIODIC SIGNAL
A periodic function is any function for which
(4)
for all t. The period T is the length of the time at which the function begins to repeat itself. Clearly the trigonometric functions sinωt and cosωt are periodic with period T=1/f=2π/ω, where f is the frequency in cycles/s (Hz) and ω is the circular (angular) frequency in radians/s. Figure 1 shows such a periodic function. Any piecewise-continuous, integrable periodic function may be represented by a superposition of sine and cosine functions
(5)
where ω0 is the fundamental frequency and ωn= nω0 is the nth harmonic of the periodic function. Equation (5) is the Fourier series representation of the periodic function y(t).
Figure 1 Example of periodic function
The orthogonality property of the sine and cosine functions gives the following expression for the Fourier coefficients an and bn:
GIBBS PHENOMENON
The well known Gibbs phenomenon represents the difficulty of (the partials sum of) Fourier series or (the truncated) Fourier integrals in approximating functions near their jump discontinuities.
In general, for well-behaved (continuous) periodic signals, a sufficiently large number of harmonics can be used to approximate the signal reasonably well. For periodic signals with discontinuities, however, such as a periodic square wave, even a large number of harmonics will not be sufficient to reproduce the square wave exactly. This effect is known as Gibbs phenomenon and it manifests itself in the form of ripples of increasing frequency and closer to the transitions of the square signal. Moreover, these ripples do not die out as the frequency increases. Figure 2 demonstrates Fourier series approximations of a square wave and Gibbs Phenomenon.
Figure 2 The Gibbs Phenomenon for truncated Fourier series of a square wave
DISCRETE FOURIER TRANSFORM (DFT)
The computational basis of classical spectrum analysis is the Discrete Fourier Transform (DFT). The digital representation of a continuous signal y(t) in the time domain is the series
The computational basis of classical spectrum analysis is the Discrete Fourier Transform (DFT). The digital representation of a continuous signal y(t) in the time domain is the series
where is the sampling interval, Srate is the sampling rate or sampling frequency, and N is the number of samples.
The DFT (Discrete Fourier Transform) of the discrete series yj is given by
(6)
where . (7)
Equation (6) performs the numerical integration corresponding to the continuous integration in the definition of the Fourier transform. The values of T(fk) represent k=N/2 discrete amplitudes spaced at discrete frequency intervals having a resolution of Srate/N. Note, the maximum frequency of the spectrum obtained from the DFT is Srate/2, i.e. the Nyquist frequency; thus there is no information on frequencies above half of the sampling rate. If the signal has content at frequencies above this value, aliasing will occur. To avoid this, signals are often filtered to remove frequency content above the Nyquist before they are sampled. Note that applying a digital filter after sampling will be ineffective, since aliasing will have already occurred.
From equation (7), it can be seen that reducing the sampling frequency Srate leads to improved frequency resolution. However, in order to avoid aliasing, the sample frequency must not be reduced to less than what is required by the Nyquist criterion. The frequency resolution may be improved without changing the sample rate by increasing the number of samples taken, but this is not always possible or practical.
The size of the data array sent to the FFT, N, must be a power of two (otherwise you have a Slow Fourier Transform). Many implementations return an array of equal size, N points, but only the first N/2 are valid. The second half will simply be a mirror image of the first half. If only N/2 points are returned, they are all valid. In addition, the frequencies are usually not returned along with the amplitude information. The frequencies can be computed knowing that the bandwidth of the spectrum is from 0 to Srate/2 Hz, and there will be N/2 equally spaced frequencies.
THE FAST FOURIER TRANSFORM (FFT). SPECTRUM ANALYSIS WITH FFT AND MATLAB
The fast Fourier transform (FFT) is simply a class of special algorithms which implement the discrete Fourier transform with considerable savings in computational time. It must be pointed out that the FFT is not a different transform from the DFT, but rather just a means of computing the DFT with a considerable reduction in the number of calculations required.
The typical syntax for computing FFT of a signal is FFT(x,N), where x – is the signal, x[n], you wish to transform, and N is the number of points in the FFT. N must be at least as large as the number of samples in x[n].
Let us generate sine curve within Matlab using the following commands:
Fs = 100; % sampling rate
Ts = 1/Fs; %sampling time interval, Time increment per sample
T=1;
t = 0:Ts:T; %time range
fo = 4; %frequency of the sine wave, 4Hz component
Fmax=1/Ts; %maximum frequency
df=1/T; %Frequency increment
f=0:df:Fmax; %Frequency range (Vector of frequency)
Length of a frequency vector: nf=length(f);
y = 2*sin(2*pi*fo*t); %the sine curve
Plot the sine curve in the time domain
figure(1)
plot(t,y),grid on
title('Sine Wave')
xlabel ('time (sec)')
ylabel ('output signal Y(t)')
Thus, we obtain
Figure 3 Sine wave signal
Let us use MATLAB’s built in fft command to try to recreate the frequency spectrum.
X=fft(y); %Discrete Fourier Transform
Ampl=abs(X); %use abs command to get the magnitude
figure(2)
stem(f,Ampl),grid on
xlabel('Sample Number')
ylabel('Amplitude')
title('Using the Matlab fft command')
Figure 4 Fourier transform of a signal
The FFT contains information between 0 and fs, however, we know that the sampling frequency must be at least twice the highest frequency component. Therefore, the signals spectrum should be entirely below fs/2, the Nyquist frequency.
Recall also, that a real signal should have a transform magnitude that is symmetrical for positive and negative frequencies. Thus, instead of having a spectrum that goes from 0 to fs, it would be more appropriate to show the spectrum from -fs/2 to fs/2. This can be accomplished by using Matlab’s fftshift function as the following code demonstrates.
fn=-Fmax/2:df:Fmax/2; % Normalized frequency axis
Xshift=fftshift(X);
AmplShift=abs(Xshift);
figure(3)
stem(fn,AmplShift),grid on
xlabel('Freq (Hz)')
ylabel('Amplitude')
title('Using the centeredFFT function')
Figure 5 Centered signal via fftshift command
In order to define Amplitude and phase of FFT signal he following syntax is used:
ReX0 = real(Xshift);
ImX0 = imag(Xshift);
The simulation results are represented below.
Figure 6 Amplitude and Phase for FFT of a signal
It is necessary to define some distinctive properties of Fourier Transform.
Firstly, if the signal y(t) is even then the following evenness conditions for spectrum Y( f ) come true:
where an asterisk defines complex conjugate component. It is seen that even signal possesses with even amplitude function, and phase is an odd function.
Secondly, if the signal is a real and even function of time
,
then the following conditions preserve truth for spectrum
Here the imaginary part of the spectrum is equal to zero.
Thirdly, if the signal is the real and odd function of time
,
Then the following conditions come true
In this case, the real part of the spectrum is equal to zero.
STANDARD TASK FOR LABORATORY WORK №3
Table 1
|
Variants |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
Shape of a finite signal |
Triangular pulse |
Rectangular pulse |
Rectangular pulse |
Triangular pulse |
Rectangular pulse |
Triangular pulse |
Rectangular pulse |
Triangular pulse |
Triangular pulse |
|
Signal property |
height 2, centered at 5 instant time, width of 0.8 |
height 6, centered at 5 instant time, width of 4 |
height 2.5, centered at 2 instant time, width of 8, |
height 5, centered at 20 instant time, width of 6, skew 0 |
height 1, centered at 9 instant time, width of 6 |
height 8, centered at 12 instant time, width of 5, skew -0.5 |
height 12, centered at 5 instant time, width of 8 |
height 4, centered at 8 instant time, width of 4 |
height 3, centered at 5 instant time, width of 10, skew 1 |
Prompt: To generate rectangular signal the following syntax is used:
rectpuls (t,w) returns a continuous, aperiodic, unity-height rectangular pulse at the sample times indicated in array t, centered about t = 0 and with a width of w, by default 1.
For triangular signal generation use command “tripuls”, namely
tripuls (T,w,s) returns a continuous, a periodic, symmetric, unity-height triangular pulse at the times indicated in array t, centered about t=0 and with a width of w (by default 1) and with skew s, where -1 < s< 1. When s is 0, a symmetric triangular pulse is generated.
Initial data for periodic signal and its type is specified in Table 2.
Table 2
|
Variants |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
Signal’s shape |
cosine |
sine |
cosine |
sine |
cosine |
sine |
cosine |
sine |
cosine |
|
f (Hz) |
10 |
25 |
40 |
20 |
10 |
30 |
20 |
30 |
40 |
|
a |
1 |
3 |
0.5 |
2 |
0.5 |
4 |
1 |
3 |
2.5 |
Define Magnitude (Power), Amplitude and Phase for the FFT of the signal. Attach the plot.
Figure 3. Simulation model
On the next stage in is necessary to adjust above given blocks. Their properties are necessary to be set as follows:
Sine Wave Source Block:
Complex to Magnitude-Angle Block: define output to Magnitude;
Vector Scope block: find Axis properties toolbar and set frequency rage to [-Fs/2…Fs/2] and Frequency Unit: Hertz;
Simulate a sine curve with amplitude 1 and frequency of 1KHz. Sampling time set to 8KHz. The output signal select to be a real. Set the size buffer to 16. Choose Window Function to be Blackman. Obtain the spectrum for different types of frequency range. Explain the obtained results.
Report
1 Submit your Matlab code.
2 Submit your calculation and hand sketch of the signal.
3 Submit the plot of magnitude (power), Amplitude and Phase of the signal.