FFT, total energy, and energy spectral density computations in MATLAB Aaron Scher

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1 FFT, total energy, and energy spectral density computations in MATLAB Aaron Scher Everything presented here is specifically focused on non-periodic signals with finite energy (also called energy signals ) Part 1 THEORY Instantaneous power of continuous-time signals: Let x t be a real (ie no imaginary part) signal If x t represents voltage across a 1 W resistor, then the instantaneous power dissipated by Ohmic losses in the resistor is: p t = % & ' ( = x t ) (Instantaneous power in [W]=[J/s]) (1) Total energy of continuous-time signals: The total energy dissipated across the resistor is the time-integral of the instantaneous power: E, = p t dt = x t ) dt (Total signal energy in [J]) (2) A signal with a finite energy E, (ie E, < ) is called an energy signal We are focusing solely on energy signals here Continuous time Fourier transform (CTFT) pair: Recall the CTFT: X f = x t e /5)67& dt (3) Energy of continuous-time signals computed in frequency domain It can be shown using Parseval s theorem that the total energy can also be computed in the frequency domain: E, = X f ) df (Total signal energy in [J] computed in frequency domain) (4) Compare equation (4) with (2) These are two ways equations to compute the total energy E Energy spectral density Looking at Equation (5), we see that we can define an energy spectral density (ESD) given by: Ψ, f = X f ) (Energy spectral density in [J/Hz]) (5) 1

2 Part 2 MATLAB Our goal in this section is to use MATLAB to plot the amplitude spectrum, energy spectral density, and numerically estimate the total energy Eg First, let s sample! How do we sample a signal x t in MATLAB? For ease, let s work specifically on an example (you can easily generalize what is presented here to other signals) Suppose our signal is a small piece of sinusoid with frequency f ; described by the function: x t = 0 t < 1 sin 2πf ; t, 1 t 1 0, t > 1 (6) where the units for time are in seconds Given f ; we need to choose a sampling frequency f G that is sufficiently high (should be higher than Nyquist rate to avoid aliasing ) For this example, let us choose f ; = 2 Hz and a sampling frequency f G = 20 Hz The MATLAB code looks like this: 1 - f0=2; %center frequency [Hz] 2 - fs=20; %sample rate [Hz] 3 - Ts=1/fs; %sample period [s] 4 - Tbegin=-1; %Our signal is nonzero over the time interval [Tbegin Tend] 5 - Tend=1; 6 - t=[tbegin:ts:tend]; %define array with sampling times 7 - x=sin(2*pi*f0*t); %define discrete-time function This is x[n] Mathematically, our discrete-time function x[n] (defined in line 7 of code above) is equal to the continuous-function x t (ignoring quantization error) at discrete points in time Hence, we may write: where t M are the sampling times defined in line 6 of code above: x[n] = x t M (7) t M = T OPQRS + n 1 T G, n = 1 N ; (8) where T G = 1 f G is the sampling period, N ; is the final index of x[n], ie N ; =length(x), and T OPQRS is the initial time where the finite-energy signal x(t) is nonzero (eg for the signal given by equation (6), T OPQRS = 1 seconds) In MATLAB, the indices n always start at 1 and are positive This can cause confusion, since in other programming languages indices commonly start at 0 So, in MATLAB, x[0] would give an error You must start with an index of 1; x[1] would be fine 2

3 If we are interested in evaluating x t M at some specific time, say at t = 0, we can figure out the index n that corresponds to t = 0, and plug it in to our expression for x However, the easier way is to simply plot our function x and look at the value of this function at t = 0, as done below 8 - figure(1) 9 - plot(t,x) 10 - grid on 11 - xlabel('time (seconds)','fontsize',14) 12 - ylabel('amplitude','fontsize',14) Amplitude Time (seconds) Figure 1 Plot of x n = x t M Discrete points are connected by straight lines and the x-axis is labeled as time in seconds This gives the illusion that this is a plot of a true continuous time signal x t It s actually a plot of a discrete time signal From the plot above, it is clear that the function x(t) = 0 at t = 0 If we weren t in the mood to plot, we could also find the index value that corresponds to t = 0, and plug that directly into x to find the value of x at t = 0, as shown in the code below: 13 - index = find(t==0,1) %find index n corresponding to t=0this returns x(index) %This returns 0, as expected Total energy approximation We can numerically estimate the total energy by approximating equation (2) with a Riemann sum: ^_ E g T G x[n] ) n=1 (Estimate total signal energy in [J]) (9) 3

4 MATLAB s FFT function Matlab s fft function is an efficient algorithm for computing the discrete Fourier transform (DFT) of a function To find the double-sided spectrum you need to use the fftshift function Equation (3) shows how to manually compute the continuous time Fourier transform (CTFT) X f of a continuous time function x t Instead of using an integral, we can use MATLAB to numerically compute the CFT at discrete frequency points f` as follows: X f` = a 7 b fftshift(fft(x,n)) (10) where N is the number of frequency points in the FFT, and f` are discrete frequency points: f` = 7 b + n 1 7 b, n = 1 N (11) ) ^/a Recall that N ; = length(x) is the number of discrete time points of the original signal x[n] In general N ; N, and you are free to choose N as large as you want (so long as your computer can handle it) In fact, you will generally set N>>N 0 A good value for N is N=2 16 The code below demonstrates how to calculate and plot the FFT 15 - N=2^16; %good general value for FFT (this is the number of discrete 16 - points in the FFT) 17 - y=fft(x,n); %compute FFT! There is a lot going on "behind the scenes" 18 - with this one line of code 19 - z=fftshift(y); %Move the zero-frequency component to the center of the 20 - %array This is used for finding the double-sided spectrum (as opposed 21 - to the single-sided spectrum) 22 - f_vec=[0:1:n-1]*fs/n-fs/2; %Create frequency vector (this is an array 23 - of numbers Each number corresponds to a discrete point in frequency 24 - in which we shall evaluate and plot the function) 25 - amplitude_spectrum=abs(z)/fs %Extract the amplitude of the 26 - spectrum Here we also apply a scaling factor of 1/fs so that 27 - the amplitude of the FFT at a frequency component equals that of the 28 - CFT and to preserve Parseval s theorem 29 - figure(2) 30 - plot(f_vec,amplitude_spectrum); 31 - set(gca,'fontsize',18) %set font size of axis tick labels to xlabel('frequency [Hz]','fontsize',18) 33 - ylabel('amplitude','fontsize',18) 34 - title('amplitude spectra','fontsize',18) 35 - grid on 36 - set(gcf,'color','w');%set background color from grey(default) to white 37 - axis tight 4

5 1 Amplitude spectra 08 Amplitude Frequency [Hz] Figure 2 Once you have computed X f` using equation (10), you can then numerically compute and plot the energy spectral density (ESD): Ψ, [f`]= X f` ) (Energy spectral density in [J/Hz]) (12) The total energy can then be found by approximating equation (4) with a Riemann sum: E g 7 b ^ ^ ) X n=1 f` (Total signal energy in [J] computed in frequency domain) (13) The code below demonstrates how to calculate and plot the energy spectral density 38 - figure(3) 39 - plot(f_vec,abs(amplitude_spectrum)^2); 40 - xlabel('frequency [Hz]','fontsize',18) 41 - ylabel('energy/hz','fontsize',18) 42 - title('energy spectral density','fontsize',18) 43 - grid on 44 - set(gcf,'color','w'); %set background color to white 45 - axis tight 1 Energy spectral density Energy/Hz Frequency [Hz] Figure 3 5

6 We can now numerically calculate the total energy using either equation (9) or (13): %calculate total energy 46 - Eg=sum(x^2)*Ts %using discrete time function x[n] 47 - Eg=sum(amplitude_spectrum^2)*fs/N %using ESD MATLAB returns Eg = 1000 for both methods The units of energy are Joules [J] The full MATLAB code for the above example is pasted below Simply save this code in an m- file and run the m-file clc; clear all; f0=2; %center frequency [Hz] fs=20; %sample rate [Hz] Ts=1/fs; %sample period [s] Tbegin=-1; %Our signal is nonzero over the time interval [Tbegin Tend] Tend=+1; t=[tbegin:ts:tend]; %define array with discrete sampling times x=sin(2*pi*f0*t); %define discrete-time function This is x[n] figure(1) plot(t,x) grid on xlabel('time (seconds)','fontsize',14) ylabel('amplitude','fontsize',14) N=2^16; %good general value for FFT (this is the number of discrete points in the FFT) y=fft(x,n); %compute FFT! There is a lot going on "behind the scenes" with %this one line of code z=fftshift(y); %Move the zero-frequency component to the center of the %array This is used for finding the double-sided spectrum (as opposed to %the single-sided spectrum) f_vec=[0:1:n-1]*fs/n-fs/2; %Create frequency vector (this is an array of %numbers Each number corresponds to a discrete point in frequency in which we %shall evaluate and plot the function) amplitude_spectrum=abs(z)/fs; %Extract the amplitude of the spectrum %Here we also apply a scaling factor of 1/fs, so that the amplitude %of the FFT at a frequency component equals that of the the CFT and to %preserve Parseval's theorem amplitude_spectrum=abs(z)*ts; figure(2) plot(f_vec,amplitude_spectrum); set(gca,'fontsize',18) %set font size of axis tick labels to 18 xlabel('frequency [Hz]','fontsize',18) ylabel('amplitude','fontsize',18) title('amplitude spectra','fontsize',18) grid on set(gcf,'color','w'); %set background color from grey (default) to white axis tight figure(3) plot(f_vec,abs(amplitude_spectrum)^2); xlabel('frequency [Hz]','fontsize',18) ylabel('energy/hz','fontsize',18) title('energy spectral density','fontsize',18) grid on set(gcf,'color','w'); %set background color from grey (default) to white axis tight %calculate total energy Eg=sum(x^2)*Ts %using discrete time function x[n] Eg=sum(amplitude_spectrum^2)*fs/N %Using ESD 6

7 Part 3 One last thing Autocorrelation Method Believe it or not, there is a second way to calculate the energy spectral density (ESD), Ψ, t Equation (5) presents one method for calculating Ψ, But folks, in life we have choices Let s examine the second method First, the time autocorrelation function is defined as: ψ, τ = x t x t + τ dt (Autocorrelation function) (14) It can be proved that the Fourier transform of the autocorrelation function is equal to the ESD, Ψ, t, ie F ψ, τ = Ψ, f = X f ) (FT of autocorrelation function is ESD) (15) In MATLAB, we can numerically compute the autocorrelation function presented in equation (14) using the xcorr function : ψ, τ M = a 7 b xcorr(x,x) (16) We can then find the ESD by taking the FFT of ψ, τ M The MATAB code below shows how to compute ESD using the autocorrelation method Figure 4 shows a spectral plot of the ESD Compare Figure 4 with Figure 3 they are identical which demonstrates the two different methods for yield identical results for the ESD %This code continues from the previous code (ie, x, Ts, fs, f_vec and %N are already defined) %Below I demonstrate an alternative method to plot EST This method %uses autocorrelation cor=ts*xcorr(x,x); %numerically compute autocorrelation %cor=ts*conv(flipud(x),x); %this is an alternative way to compute %autocorrelation y=fft(cor,n)/fs; %take FFT of autocorrrelation function and scale 7

8 %This yields the ESD z=fftshift(y); %we want to plot double-sided spectrum figure(4) %open new figure plot(f_vec,abs(z)) %plot ESD xlabel('frequency [Hz]','fontsize',18) ylabel('energy/hz','fontsize',18) title('energy spectral density - Via FFT of Autocorrelation','fontsize',18) grid on set(gcf,'color','w'); %set background color from grey (default) to white axis tight %calculate total energy with autocorrelation function Eg=sum(abs(z))*fs/N %Using ESD Energy spectral density - Via FFT of Autocorrelation Energy/Hz Frequency [Hz] Figure 4 8