Fast Fourier Transform or FFT is a powerful tool to visualize a signal in the frequency domain. Shown below is the FFT of a signal (press the play button) composed of four sinusoids at frequencies of 50Hz, 100Hz, 200Hz and 400Hz. The sampling frequency is set at 1000Hz, more than twice the maximum frequency of the composite signal. The amplitude of the sinusoids diminishes with increasing frequency and this is also reflected in the frequency domain. Play with the code, decrease the frequencies, increase the power, visualize the FFT output on logarithmic scale!
Most of us have used the FFT routine in MATLAB. This routine has become increasingly important in simulation of communication systems as it is being used in Orthogonal Frequency Division Multiplexing (OFDM) which is employed in 4G technologies like LTE and WiMAX. We would not go into the theoretical details of the FFT, rather, we would produce the MATLAB code for it and leave the theoretical discussion for a later time.
The underlying technique of the FFT algorithm is to divide a big problem into several smaller problems which are much easier to solve and then combine the results in the end.
%%%%% INITIALIZATION %%%%% clear all close all fm=100; fs=1000; Ts=1/fs; nn=1024; isign=1; t=0:Ts:(nn-1)*Ts; x_c=exp(1i*2*pi*fm*t); x(1:2:2*nn-1)=imag(x_c); x(2:2:2*nn)=real(x_c); %%%%% BIT REVERSAL %%%%% n=2*nn; j=1; for i=1:2:n-1 if(j>i) temp=x(j); x(j)=x(i); x(i)=temp; temp=x(j+1); x(j+1)=x(i+1); x(i+1)=temp; end m=nn; while (m >= 2 && j > m) j=j-m; m=m/2; end j=j+m; end %%%%% DANIELSON-LANCZOS ALGO %%%%% mmax=2; while (n > mmax) istep=2*mmax; theta=isign*(2*pi/mmax); wtemp=sin(0.5*theta); wpr=-2*wtemp*wtemp; wpi=sin(theta); wr=1.0; wi=0.0; for m=1:2:mmax-1 for i=m:istep:n j=i+mmax; tempr=wr*x(j)-wi*x(j+1); tempi=wr*x(j+1)+wi*x(j); x(j)=x(i)-tempr; x(j+1)=x(i+1)-tempi; x(i)=x(i)+tempr; x(i+1)=x(i+1)+tempi; end wtemp=wr; wr=wr*wpr-wi*wpi+wr; wi=wi*wpr+wtemp*wpi+wi; end mmax=istep; end F=x(2:2:n)+1i*x(1:2:n-1); plot((0:fs/(nn-1):fs),abs(F))
In the above example we have calculated an ‘nn’ point complex FFT of an ‘nn’ point complex time domain signal. The algorithm takes the input in a special arrangement where the ‘nn’ point complex input signal is converted into ‘2*nn’ real sequence where the imaginary components are placed in odd elements and real components are placed in even elements of the input sequence. A similar arrangement works for the output sequence.
Shown below is the FFT of a complex exponential with a frequency of 100 Hz. The plot is shown from 0Hz to 1000 Hz which is the sampling frequency. A signal with multiple frequencies would have to be passed through a Low Pass Filter (LPF) so that the signal components above 500 Hz (fs/2) are filtered out. When the FFT of a real signal is performed an image frequency is produced between 500 Hz to 1000 Hz.
Here we have discussed the case of complex input sequence. Simplifications can be made for a real sequence or for special signals such as pure sine and cosine waves. We will discuss these in later posts.
 Numerical Recipes in C, The Art of Scientific Computing, Cambridge University Press.
We have previously discussed the problem of detecting two closely spaced sinusoids using the Discrete Fourier Transform (DFT). We assumed that the data set we got was pure i.e. there was no noise. However, in reality this is seldom the case. There is always some noise, corrupting the signal. Let us now see how it effects the detection problem.
We consider Additive White Gaussian Noise (AWGN) as the corrupting source. The noise power is set equal to the power of the two sinusoids i.e. we have an SNR of 0 dB. This is quite a severe case, the noise power is usually a few dB below the signal power. We are also bounded by the number of samples, N=64, giving us a resolution of 15.87 Hz.
clear all close all fm1=100; fm2=120; fs=1000; Ts=1/fs; N=64; t=0:Ts:(N-1)*Ts; x1=sqrt(2)*cos(2*pi*fm1*t); x2=sqrt(2)*cos(2*pi*fm2*t); wn=randn(1,N); x=x1+x2+wn; W=exp(-j*2*pi/N); n=0:N-1; k=0:N-1; X=x*(W.^(n'*k)); plot(k/N,20*log10(abs(X))) xlabel ('Normalized Frequency') ylabel ('X(dB)')
The data is plotted on a logarithmic scale so that we can compare the signal and noise power levels.
It is observed that we still have two peaks around the required frequency bins but there are also a number of false peaks. These peaks are around 10 dB lower than the signal peaks and should not cause a false detection. So the signal to noise ratio of 0 dB in the time domain is translated to a signal to noise ratio of about 10 dB in the frequency domain (this can be realized using an appropriate filter).