I have a time varying signal (time,amplitude) and a measured frequency sensitivity (frequency,amplitude conversion factor (Mf)).
I know that if I use the center frequency of my time signal to select the amplitude conversion factor (e.g. 0.0312) for my signal I get a max. converted amplitude value of 1.4383.
I have written some code to deconvolve the time varying signal and known sensitivity (i.e. for all frequencies).
where Pt is the output/converted amplitude and Mf is amplitude conversion factor data and fft(a) is the fft of the time varying signal (a).
I take the real part of the fft(a):
xdft = fft(a);
xdft = xdft(1:length(x)/2+1); % only retaining the positive frequencies
freq = Fs*(0:(L/2))/L;
where Fs is sampling frequency and L is length of signal.
convS = real(xdft).*Mf;
assuming Mf is magnitude = real (I don't have phase info). I also interpolate
Mf=interp1(freq_Mf,Mf_in,freq,'cubic');
so at the same interrogation points as freq.
I then reconstruct the signal in time domain using:
fftRespI=complex(real(convS),imag(xdft));
pt = ifft(fftRespI,L,'symmetric')
where I use the imaginary part of the fft(a).
The reconstructed signal shape looks correct but the amplitude of the signal is not.
If I set all values of Mf = 0.0312 for f=0..N MHz I expect a max. converted amplitude value of ~1.4383 (similar to if I use the center frequency) but I get 13.0560.
How do I calibrate the amplitude axis? i.e. how do I correctly multiply fft(a) by Mf?
Some improved understanding of the y axis of the abs(magnitude) and real FFT would help me I think...
thanks
You need to re-arrange the order of your weights Mf to match the MATLAB's order for the frequencies. Let's say you have a signal
N=10;
x=randn(1,N);
y=fft(x);
The order of the frequencies in the output y is
[0:1:floor(N/2)-1,floor(-N/2):1:-1] = [0 1 2 3 4 -5 -4 -3 -2 -1]
So if your weights
Mf = randn(1,N)
are defined in this order: Mf == [0 1 2 3 4 5 6 7 8 9], you will have to re-arrange
Mfshift = [Mf(1:N/2), fliplr(Mf(2:(N/2+1)))];
and then you can get your filtered output
z = ifft(fft(x).*Mshift);
which should come out real.
Related
I have a hard time wrapping my head around this since there is non-linearity involved. How do I convert a spectrum I have in units "1/m" (or cycles per m) to "cycles/deg"?
Example: For simplicity let's work in one dimension and the screen of a laptop is 768 pixels on which I show the highest pixel frequency. I use MATLAB/Octave style code but I hope it's clear enough to act as pseudo code:
X = 768
x = 0:X-1
y = cos(2*pi/X*(X/2)*x)
The frequency in this signal is now pi radiants or 0.5 cycles/pixel (sampling rate is 1 pixel). I can create the frequency vectors:
w = linspace(0, 2*pi, length(x)) % in radiants
f = linspace(0, 1, length(x)) % in 1/pixel or cycles/pixel
I can easily convert this to spatial frequency in units [m] and frequency [cycles/m] when I know the pixel size. Let's assume one pixel is 0.5mm:
xm = x*0.5e-3 % for image in m
fm = f/0.5e-3 % frequency in 1/m
Plotting y versus xm shows now a waveform with 1 cycle per mm. Plotting the FFT of y versus fm shows a peak at 1000 cycles/m (or 1 cycle per mm), as expected.
How do I need to modify fm to arrive at fdeg, in other words, to show the FFT of y in cycles per degree, assuming distance from the screen is 60cm? I would assume
fdeg = 1/(2*atan(1/fm)/(2*0.6)*180/pi)
That would give me 20.97 cycles/deg. However, using the small angle approximation, 1mm should be roughly 1mm/0.6*180/pi=0.9549e-3 degree, so we have 1cycle per 0.9549e-3 degrees or 1/0.9549e-3=10.47 cycles per degree. This is off by a factor of 2.
Considering an Additive White Gaussian Noise (AWGN) communication channel where a signal taking values from BPSK modulation is being transmitted. Then, the received noisy signal is :y[k] = s[k] + w[k] where s[k] is either +1,-1 symbol and w[k] is the zero mean white gaussian noise.
-- I want to estimate the signal s and evaluate the performance by varing SNR from 0:40 dB. Let, the estimated signal be hat_s.
So, the graph for this would have on X axis the SNR range and on Y Axis the Mean Square Error obtained between the known signal values and the estimates i.e., s[k] - hat_s[k]
Question 1: How do I define signal-to-noise ratio? Would the formula of SNR be sigma^2/sigma^2_w. I am confused about the term in the numerator: what is the variance of the signal, sigma^2, usually considered?
Question 2: But, I don't know what the value of the variance of the noise is, so how does one add noise?
This is what I have done.
N = 100; %number of samples
s = 2*round(rand(N,1))-1;
%bpsk modulation
y = awgn(s,10,'measured'); %adding noise but I don't know
the variance of the signal and noise
%estimation using least squares
hat_s = y./s;
mse_s = ((s-hat_s).^2)/N;
Please correct me where wrong. Thank you.
First I think that it is important to know what are the things we have an a BPSK system:
The constellation of a BPSK system is [-A , A] in this case [-1,1]
the SNR will vary from 0 db to 40 db
I thing that the answer is in this function:
y = awgn( ... ); from matlab central:
y = awgn(x,snr) adds white Gaussian noise to the vector signal x. The
scalar snr specifies the signal-to-noise ratio per sample, in dB. If x
is complex, awgn adds complex noise. This syntax assumes that the
power of x is 0 dBW.
y = awgn(x,snr,sigpower) is the same as the syntax above, except that
sigpower is the power of x in dBW.
y = awgn(x,snr,'measured') is the same as y = awgn(x,snr), except that
awgn measures the power of x before adding noise.
you use y = awgn(x,snr,'measured'), so you do not need to worry, beacuse matlab carries all for you, measure the power of the signal, and then apply to channel a noise with the variance needed to get that SNR ratio.
let's see how can this happen
SNRbit = Eb/No = A^2/No = dmin^2 /4N0
the constelation [A,-A] in this case is [-1,1] so
10 log10(A^2/N0) = 10 log10(1/N0) = SNRbitdb
SNRlineal = 10^(0.1*SNRdb)
so with that:
noise_var=0.5/(EbN0_lin); % s^2=N0/2
and the signal will be something like this
y = s + sqrt(noise_var)*randn(1,size);
so in your case, I will generate the signal as you do:
>> N = 100; %number of samples
>> s = 2*round(rand(N,1))-1; %bpsk modulation
then prepare a SNR varies from 0 to 40 db
>> SNR_DB = 0:1:40;
after that calulating all the posible signals:
>> y = zeros(100,length(SNR_DB));
>> for i = 1:41
y(:,i) = awgn(s,SNR_DB(i),'measured');
end
at this point the best way to see the signal is using a constellation plot like this:
>> scatterplot(y(:,1));
>> scatterplot(y(:,41));
you can see a bad signal 0 db noise equal power as signal and a very good signal signal bigger than 40 DB noise. Eb/No = Power signal - Power noise db, so 0 means power noise equal to power of signal, 40 db means power of signal bigger bigger bigger than power of noise
then for you plot calculate the mse, matlab has one function for this
err = immse(X,Y) Description
example
err = immse(X,Y) calculates the mean-squared error (MSE) between the
arrays X and Y. X and Y can be arrays of any dimension, but must be of
the same size and class.
so with This:
>> for i = 1:41
err(i) = immse(s,y(:,i));
end
>> stem(SNR_DB,err)
For plots, and since we are working in db, it should be beeter to use logarithmic axes
I would like to create 500 ms of band-limited (100-640 Hz) white Gaussian noise with a (relatively) flat frequency spectrum. The noise should be normally distributed with mean = ~0 and 99.7% of values between ± 2 (i.e. standard deviation = 2/3). My sample rate is 1280 Hz; thus, a new amplitude is generated for each frame.
duration = 500e-3;
rate = 1280;
amplitude = 2;
npoints = duration * rate;
noise = (amplitude/3)* randn( 1, npoints );
% Gaus distributed white noise; mean = ~0; 99.7% of amplitudes between ± 2.
time = (0:npoints-1) / rate
Could somebody please show me how to filter the signal for the desired result (i.e. 100-640 Hz)? In addition, I was hoping somebody could also show me how to generate a graph to illustrate that the frequency spectrum is indeed flat.
I intend on importing the waveform to Signal (CED) to output as a form of transcranial electrical stimulation.
The following is Matlab implementation of the method alluded to by "Some Guy" in a comment to your question.
% In frequency domain, white noise has constant amplitude but uniformly
% distributed random phase. We generate this here. Only half of the
% samples are generated here, the rest are computed later using the complex
% conjugate symmetry property of the FFT (of real signals).
X = [1; exp(i*2*pi*rand(npoints/2-1,1)); 1]; % X(1) and X(NFFT/2) must be real
% Identify the locations of frequency bins. These will be used to zero out
% the elements of X that are not in the desired band
freqbins = (0:npoints/2)'/npoints*rate;
% Zero out the frequency components outside the desired band
X(find((freqbins < 100) | (freqbins > 640))) = 0;
% Use the complex conjugate symmetry property of the FFT (for real signals) to
% generate the other half of the frequency-domain signal
X = [X; conj(flipud(X(2:end-1)))];
% IFFT to convert to time-domain
noise = real(ifft(X));
% Normalize such that 99.7% of the times signal lies between ±2
noise = 2*noise/prctile(noise, 99.7);
Statistical analysis of around a million samples generated using this method results in the following spectrum and distribution:
Firstly, the spectrum (using Welch method) is, as expected, flat in the band of interest:
Also, the distribution, estimated using histogram of the signal, matches the Gaussian PDF quite well.
I'm working on sound signals of a walking pattern, which has obvious regular patterns:
Then I thought I can get the frequency of walking (approximately 1.7Hz from the image) using FFT function:
x = walk_5; % Walking sound with a size of 711680x2 double
Fs = 48000; % sound frquency
L=length(x);
t=(1:L)/Fs; %time base
plot(t,x);
figure;
NFFT=2^nextpow2(L);
X=fft(x,NFFT);
Px=X.*conj(X)/(NFFT*L); %Power of each freq components
fVals=Fs*(0:NFFT/2-1)/NFFT;
plot(fVals,Px(1:NFFT/2),'b','LineSmoothing','on','LineWidth',1);
title('One Sided Power Spectral Density');
xlabel('Frequency (Hz)')
ylabel('PSD');
But then it doesn't give me what I expected:
FFT result:
zoom image has lots of noises:
and there is no information near 1.7Hz
Here is the graph from log domain using
semilogy(fVals,Px(1:NFFT));
It's pretty symmetric though:
I couldn't find anything wrong with my code. Do you have any solutions to easily extract the 1.7Hz from the walking pattern?
here is the link for the audio file in mat
https://www.dropbox.com/s/craof8qkz9n5dr1/walk_sound.mat?dl=0
Thank you very much!
Kai
I suggest you to forget about DFT approach since your signal is not appropriate for this type of analysis due to many reasons. Even by looking on the spectrum in range of frequencies that you are interested in, there is no easy way to estimate the peak:
Of course you could try with PSD/STFT and other funky methods, but this is an overkill. I can think of two, rather simple methods, for this task.
First one is based simply on the Auto Correlation Function.
Calculate the ACF
Define the minimum distance between them. Since you know that expected frequency is around 1.7Hz, then it corresponds to 0.58s. Let's make it 0.5s as the minimum distance.
Calculate the average distance between peaks found.
This gave me an approximate frequency of 1.72 Hz .
Second approach is based on the observation to your signal already has some peaks which are periodic. Therefore we can simply search for them using findpeaks function.
Define the minimum peak distance in a same way as before.
Define the minimum peak height. For example 10% of maximum peak.
Get the average difference.
This gave me an average frequency of 1.7 Hz.
Easy and fast method. There are obviously some things that can be improved, such as:
Refining thresholds
Finding both positive and negative peaks
Taking care of some missing peaks, i.e. due to low amplitude
Anyway that should get you started, instead of being stuck with crappy FFT and lazy semilogx.
Code snippet:
load walk_sound
fs = 48000;
dt = 1/fs;
x = walk_5(:,1);
x = x - mean(x);
N = length(x);
t = 0:dt:(N-1)*dt;
% FFT based
win = hamming(N);
X = abs(fft(x.*win));
X = 2*X(1:N/2+1)/sum(win);
X = 20*log10(X/max(abs(X)));
f = 0:fs/N:fs/2;
subplot(2,1,1)
plot(t, x)
grid on
xlabel('t [s]')
ylabel('A')
title('Time domain signal')
subplot(2,1,2)
plot(f, X)
grid on
xlabel('f [Hz]')
ylabel('A [dB]')
title('Signal Spectrum')
% Autocorrelation
[ac, lag] = xcorr(x);
min_dist = ceil(0.5*fs);
[pks, loc] = findpeaks(ac, 'MinPeakDistance', min_dist);
% Average distance/frequency
avg_dt = mean(gradient(loc))*dt;
avg_f = 1/avg_dt;
figure
plot(lag*dt, ac);
hold on
grid on
plot(lag(loc)*dt, pks, 'xr')
title(sprintf('ACF - Average frequency: %.2f Hz', avg_f))
% Simple peak finding in time domain
[pkst, loct] = findpeaks(x, 'MinPeakDistance', min_dist, ...
'MinPeakHeight', 0.1*max(x));
avg_dt2 = mean(gradient(loct))*dt;
avg_f2 = 1/avg_dt2;
figure
plot(t, x)
grid on
hold on
plot(loct*dt, pkst, 'xr')
xlabel('t [s]')
ylabel('A')
title(sprintf('Peak search in time domain - Average frequency: %.2f Hz', avg_f2))
Here's a nifty solution:
Take the absolute value of your raw data before taking the FFT. The data has a ton of high frequency noise that is drowning out whatever low frequency periodicity is present in the signal. The amplitude of the high frequency noise gets bigger every 1.7 seconds, and the increase in amplitude is visible to the eye, and periodic, but when you multiply the signal by a low frequency sine wave and sum everything you still end up with something close to zero. Taking the absolute value changes this, making those amplitude modulations periodic at low frequencies.
Try the following code comparing the FFT of the regular data with the FFT of abs(data). Note that I took a few liberties with your code, such as combining what I assume were the two stereo channels into a single mono channel.
x = (walk_5(:,1)+walk_5(:,2))/2; % Convert from sterio to mono
Fs = 48000; % sampling frquency
L=length(x); % length of sample
fVals=(0:L-1)*(Fs/L); % frequency range for FFT
walk5abs=abs(x); % Take the absolute value of the raw data
Xold=abs(fft(x)); % FFT of the data (abs in Matlab takes complex magnitude)
Xnew=abs(fft(walk5abs-mean(walk5abs))); % FFT of the absolute value of the data, with average value subtracted
figure;
plot(fVals,Xold/max(Xold),'r',fVals,Xnew/max(Xnew),'b')
axis([0 10 0 1])
legend('old method','new method')
[~,maxInd]=max(Xnew); % Index of maximum value of FFT
walkingFrequency=fVals(maxInd) % print max value
And plotting the FFT for both the old method and the new, from 0 to 10 Hz gives:
As you can see it detects a peak at about 1.686 Hz, and for this data, that's the highest peak in the FFT spectrum.
I'm trying to find the peak frequency for two signals 'CA1' and 'PFC', within a specified range (25-140Hz).
In Matlab, so far I have plotted an FFT for each of these signals (see pictures below). These FFTs suggest that the peak frequency between 25-140Hz is different for each signal, but I would like to quantify this (e.g. CA1 peaks at 80Hz, whereas PFC peaks at 55Hz). However, I think the FFT is not smooth enough, so when I try and extract the peak frequencies it doesn't make sense as my code pulls out loads of values. I was only expecting a few values - one each time the FFT peaks (around 2Hz, 5Hz and ~60Hz).
I want to know, between 25-140Hz, what is the peak frequency in 'CA1' compared with 'PFC'. 'CA1' and 'PFC' are both 152401 x 7 matrices of EEG data, recorded
from 7 separate individuals. I want the MEAN peak frequency for each data set (i.e. averaged across the 7 test subjects for CA1 and PFC).
My code so far (based on Matlab help files and code I've scrabbled together online):
Fs = 508;
%notch filter
[b50,a50] = iirnotch(50/(Fs/2), (50/(Fs/2))/70);
CA1 = filtfilt(b50,a50,CA1);
PFC = filtfilt(b50,a50,PFC);
%FFT
L = length(CA1);
NFFT = 2^nextpow2(L);
%FFT for each of the 7 subjects
for i = 1:size(CA1,2);
CA1_FFT(:,i) = fft(CA1(:,i),NFFT)/L;
PFC_FFT(:,i) = fft(PFC(:,i),NFFT)/L;
end
%Average FFT across all 7 subjects - CA1
Mean_CA1_FFT = mean(CA1_FFT,2);
% Mean_CA1_FFT_abs = 2*abs(Mean_CA1_FFT(1:NFFT/2+1));
%Average FFT across all 7 subjects - PFC
Mean_PFC_FFT = mean(PFC_FFT,2);
% Mean_PFC_FFT_abs = 2*abs(Mean_PFC_FFT(1:NFFT/2+1));
f = Fs/2*linspace(0,1,NFFT/2+1);
%LEFT HAND SIDE FIGURE
plot(f,2*abs(Mean_CA1_FFT(1:NFFT/2+1)),'r');
set(gca,'ylim', [0 2]);
set(gca,'xlim', [0 200]);
[C,cInd] = sort(2*abs(Mean_CA1_FFT(1:NFFT/2+1)));
CFloor = 0.1; %CFloor is the minimum amplitude value (ignore small values)
Amplitudes_CA1 = C(C>=CFloor); %find all amplitudes above the CFloor
Frequencies_CA1 = f(cInd(1+end-numel(Amplitudes_CA1):end)); %frequency of the peaks
%RIGHT HAND SIDE FIGURE
figure;plot(f,2*abs(Mean_PFC_FFT(1:NFFT/2+1)),'r');
set(gca,'ylim', [0 2]);
set(gca,'xlim', [0 200]);
[P,pInd] = sort(2*abs(Mean_PFC_FFT(1:NFFT/2+1)));
PFloor = 0.1; %PFloor is the minimum amplitude value (ignore small values)
Amplitudes_PFC = P(P>=PFloor); %find all amplitudes above the PFloor
Frequencies_PFC = f(pInd(1+end-numel(Amplitudes_PFC):end)); %frequency of the peaks
Please help!! How do I calculate the 'major' peak frequencies from an FFT, and ignore all the 'minor' peaks (because the FFT is not smoothed).
FFTs assume that the signal has no trend (this is called a stationary signal), if it does then this will give a dominant frequency component at 0Hz as you have here. Try using the MATLAB function detrend, you may find this solves your problem.
Something along the lines of:
x = x - mean(x)
y = detrend(x, 'constant')