I am trying to generate Gussin white noise with standard deviation 7% and sample frequency 2000 Hz. Then calculate the power spectral density of it (PSD). The problem is when I plotted PSD I know that I have to get flat PSD. however, I did not
I used below code
clear;
clc;
x =7*randn(1,10000);
N=10000;
sample_frequency=2000;
fax_bin=[0 :N/2-1];
fax_Hz=fax_bin*sample_frequency/N;
FFT_A=fft(x);
spectra=FFT_A.*conj(FFT_A);
figure
plot(fax_Hz,spectra(1,1:5000));
It is a random process so you will never get it truly flat, it gets better with more samples. Additionally, plotting it on a dB-scale makes more sense.
plot(fax_Hz,10*log10(spectra(1,1:N/2)));
Related
I'm trying to plot a simple signal in fourier domain using Matlab. It's not plotting the correct signal. Here is my code:
clc;
clear all;
close all;
x=1:0.001:10;
f1=sin(2*pi*10*x);
f2=sin(2*pi*15*x);
f3=sin(2*pi*30*x);
f=f1+f2+f3;
plot(2*pi*x,fft(f1));
figure
plot(x,fft(f1));
I've expected a peak at 10 since the frequency is 10. But it is giving a peak at some other point
Here are the two plot images:
This is the image for plot(x,fft(f1))
This is the image for plot(2*pi*x,fft(f1))
It is not showing the peak at 10.I even tried using abs(fft(f1)). No luck :/
Isn't it the correct way to plot signal in fourier domain?
The fft function assumes unit time step. In order to correct for non unit time step you need to define the frequency component based on the nyquist rate. The following code plots the magnitude of the fft with the correct frequency axis.
clc;
clear all;
close all;
x=1:0.001:10;
% ^ this is your sampling time step
f1=sin(2*pi*10*x);
f2=sin(2*pi*15*x);
f3=sin(2*pi*30*x);
% bounds of fourier transform based on sampling rate
Fs = 1/0.001;
ff = linspace(-Fs/2,Fs/2,numel(x));
F1 = fftshift(fft(f1)/numel(x));
F2 = fftshift(fft(f2)/numel(x));
F3 = fftshift(fft(f3)/numel(x));
figure();
plot(ff,abs(F1),'-r'); hold on;
plot(ff,abs(F2),'-b');
plot(ff,abs(F3),'-k');
Edit: To answer OPs question in the comment.
Speaking in normalized frequency units (assuming sampling rate of 1). The fft function returns the frequency response from 0 to 2*pi radians, but due to some signal processing properties and the way that discrete signals are interpreted when performing an FFT, the signal is actually periodic so the pi to 2*pi section is identical to the -pi to 0 section. To display the plot with the DC component (0 frequency) in the center we use fftshift which does a circular shift equal to 1/2 the length of the signal on the data returned by fft. Before you take the ifft make sure you use ifftshift to put it back in the right place.
Edit2: The normalization term (/numel(x)) is necessary to estimate the continuous time fourier transform using the discrete fourier transform. I don't remember the precise mathematical reason off the top of my head but the examples in the MATLAB documentation also imply the necessity of this normalization.
Edit 3: The original link that I had is down. I may come back to add a more detailed answer but in the mean time I definitely recommend that anyone interested in understanding the relationship between the fundamentals of the FS, FT, DTFT, and DFT watch Professor Oppenheim's hilariously old, but amazingly informative and straightforward lectures on MIT OpenCourseWare.
I am currently working on a project for my Speech Processing course and have just finished making a time waveform plot as well as both wide/narrow band spectrograms for a spoken word in Spanish (aire).
The next part of the project is as follows:
Make a 3-D plot of each word signal, as a function of time, frequency and power spectral density. The analysis time step should be 20ms, and power density should be computed using a 75%-overlapped Hamming window and the FFT. Choose a viewing angle that best highlights the signal features as they change in time and frequency.
I was hoping that someone can offer me some guidance as to how to begin doing this part. I have started by looking here under the Spectrogram and Instantaneous Frequency heading but was unsure of how to add PSD to the script.
Thanks
I am going to give you an example.
I am going to generate a linear chirp signal.
Fs = 1000;
t = 0:1/Fs:2;
y = chirp(t,100,2,300,'linear');
And then, I am going to define number of fft and hamming window.
nfft=128;
win=hamming(nfft);
And then I am going to define length of overlap, 75% of nfft.
nOvl=nfft*0.75;
And then, I am performing STFT by using spectrogram function.
[s,f,t,pxx] = spectrogram(y,win,nOvl,nfft,Fs,'psd');
'y' is time signal, 'win' is defined hamming window, 'nOvl' is number of overlap, 'nfft' is number of fft, 'Fs' is sampling frequency, and 'psd' makes the result,pxx, as power spectral density.
Finally, I am going to plot the 'pxx' by using waterfall graph.
waterfall(f,t,pxx')
xlabel('frequency(Hz)')
ylabel('time(sec)')
zlabel('PSD')
The length of FFT, corresponding to 20ms, depends on sampling frequency of your signal.
EDIT : In plotting waterfall graph, I transposed pxx to change t and f axis.
I am working with biological signal data, and am trying to count the number of regions with a high density of high amplitude peaks. As seen in the figure below, the regions of interest (as observed qualitatively) are contained in red boxes and 8 such regions were observed for this particular trial. The goal is to mathematically achieve this same result in near real time without the intervention or observation of the researcher.
The data seen plotted below is the result of raw data from a 24-bit ADC being processed by an FIR filter, with no other processing yet being done.
What I am looking for is a method, or ideally code, to help me detect such regions as identified while subsequently ignoring some of the higher amplitude peaks in between the regions of interest (i.e. between regions 3 and 4, 5 and 6, or 7 and 8 there is a narrow region of high amplitude which is not of concern). It is worth noting that the maximum is not known prior to computation.
Thanks for your help.
Data
https://www.dropbox.com/s/oejyy6tpf5iti3j/FIRData.mat
can you work with thresholds?
define:
(1) "amplitude threshold": if the signal is greater than the threshold it is considered a peak
(2) "window size" : of a fixed time duration
algorithm:
if n number of peaks was detected in a duration defined in "window size" than consider the signal within "window size" as cluster of peaks.(I worked with eye blink eeg data this way before, not sure if it is suitable for your application)
P.S. if you have data that are already labelled by human, you can train a classifier to find out your thresholds and window size.
Does it make sense in your problem to have some sort of "window size"? In other words, given a region of "high" amplitude, if you shrink the duration of the region, at what point will it become meaningless to your analysis?
If you can come up with a window, just apply this window to your data as it comes in and compute the energy within the window. Then, you can define some energy threshold and perform simple peak detection on the energy signal.
By inspection of your data, the regions with high amplitude peaks are repeated at what appears to be fairly uniform intervals. This suggests that you might fit a sine or cosine wave (or a combination of the two) to your data.
Excuse my crude sketch but what I mean is something like this:
Once you make this identification, you can use the FFT to get the dominant spatial frequencies. Keep in mind that the spatial frequency spectrum of your signal may be fairly complex, due to spurious data, but what you are after is one or two dominant frequencies of your data.
For example, I made up a sinusoid and you can do the calculation like this:
N = 255; % # of samples
x = linspace(-1/2, 1/2, N);
dx = x(2)-x(1);
nu = 8; % frequency in cycles/interval
vx = (1/(dx))*[-(N-1)/2:(N-1)/2]/N; % spatial frequency
y = sin(2*pi*nu*x); % this would be your data
F = fftshift(abs(fft(y))/N);
figure; set(gcf,'Color',[1 1 1]);
subplot(2,1,1);plot(x,y,'-b.'); grid on; xlabel('x'); grid on;
subplot(2,1,2);plot(vx,F,'-k.'); axis([-1.3*nu 1.3*nu 0 0.6]); xlabel('frequency'); grid on;
Which gives:
Note the peaks at ± nu, the dominant spatial frequency. Now once you have the dominant spatial frequencies you can reconstruct the sine wave using the frequencies that you have obtained from the FFT.
Finally, once you have your sine wave you can identify the boxes with centers at the peaks of the sine waves.
This is also a nice approach because it effectively filters out the spurious or less relevant spikes, helping you to properly place the boxes at your intended locations.
Since I don't have your data, I wasn't able to complete all of the code for you, but the idea is sound and you should be able to proceed from this point.
making pink noise (1/f) using list of frequencies
I would like to see what type of noise I would get if I used just the frequency in my voice. I created a matlab/octave array using fft to get the [frequency,amplitude,phase] to reproduce my vocal signal.
I would like to take this file/data and use it to create pink noise (1/f). Of course when I use 1/f for the frequency the numbers become very small does anyone have any ideas how to use my own vocal frequencies that I get from doing a fft in matlab to create pink noise (1/f).
Thanks
If I am correct, what you are doing is generating noise based on a 1/f frequency. However if you read the following article: http://en.wikipedia.org/wiki/Pink_noise the frequencies are the same except that the power spectral density is S is proportional to 1/f. Hence you should not be generating noise of frequency 1/f.
I would suggest reading this page for the necessary algorithms.
However if the problem you are facing is that the volume is too low, try amplifying the synthesized noise by multiplying the result by a factor ex.: pinkNoise = pinkNoise * 100
This might do the trick: compute the mean power in your spectrum from the amplitude A = A(f), where f is the frequency.
P = mean(A.^2);
Spread that over your frequency range:
N = length(f);
invfnorm = 1./[1:N];
Anew = sqrt(P*invfnorm/sum(invfnorm));
Anew has the property of having the same total power density as the original spectrum, and power decaying as 1/f.
Substitute Anew for A and inverse FFT your new spectrum to generate the new waveform.
I am trying to prove that the white noise has constant power spectral density using matlab
but the amplitude of the spectrum looks like random amplitude.
can anyone tell me why?
here is my code.
noise = randn(1,10000);
fft_noise=fft(noise);
plot(abs(fft_noise(1:5000)))
thanks.
You need to average a bunch (law of large numbers) of FFTs of white noise to approach the average power spectral density.
If you take the FFT of an independent set of random variables from the same distribution, then you'll get an independent set of random variables from the same distribution since the inverse Fourier transform is (more or less) the same as the Fourier transform. The point is that the expected value for each frequency is the same.
you need to multiple the fft by the complex conjugate of the fft to show a flat PSD. i.e. change
fft_noise=fft(noise);
to
fft_noise=fft(noise).*conj(fft(noise));