FFT – RAYmaps

December 22, 2022December 9, 2024Fundamentals

Demystifying the Fourier Transform

The Fourier Transform is often used in Communication and Signal Processing to find the spectral content of a time-domain signal. The most common example is that of a sinusoid in the time domain, resulting in a sharply peaked signal in the frequency domain, also known as a delta function. A rectangular pulse in the time domain has a more complicated frequency domain equivalent, a sinc function. A rectangular pulse may be thought of as a combination of many sinusoids, hence its frequency domain equivalent is not that straightforward.

July 3, 2020December 9, 2024Fundamentals

A Comparison of FFT, MUSIC and ESPRIT Methods of Frequency Estimation

As discussed in previous posts it is frequently required in communications and signal processing to estimate the frequency of a signal embedded in noise and interference. The problem becomes more complicated when the number of observations (samples) is quite limited. Typically, the resolution in the frequency domain is inversely proportional to the window size in the time domain.

June 17, 2020June 3, 2022Fundamentals

Frequency Estimation Using Zero Crossing Method

In this post we present the simplest method of frequency estimation that is called the Zero Crossing (ZC) method. Since a sine wave crosses the x-axis twice during each cycle, we can simply count the number of crossings and divide it by two and again divide it by the observation window size, giving us the frequency in Hertz.

August 3, 2017August 3, 2017Fundamentals

Fourier Transform in Python

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!

May 16, 2012May 16, 2012Fundamentals

Fast Fourier Transform – Code

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 […]

May 2, 2012May 2, 2012Fundamentals

Detecting Sinusoids in Noise

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 […]

April 30, 2012April 30, 2012Fundamentals

Resolution of Discrete Fourier Transform

In the previous post we had introduced the Discrete Fourier Transform (DFT) as a method to perform the spectral analysis of a time domain signal. We now discuss an important property of the DFT, its spectral resolution i.e. its ability to resolve two signals with similar spectral content. Initially one might think that increasing the sampling frequency would increase the spectral resolution but this totally incorrect. In fact if the sampling frequency is increased, keeping the number of time domain samples to be the same, the resolution actually decreases. So how do we calculate the spectral resolution. One simple way is […]

April 27, 2012April 29, 2012Fundamentals

Discrete Fourier Transform

Discrete Fourier Transform or DFT is a mathematical operation that transforms a time domain signal to frequency domain. It is usually implemented using the Fast Fourier Transform (FFT). The computational complexity of the DFT is N2 whereas its (N)log2N for the FFT, where N is the number of samples of the the time domain signal. Mathematically, the DFT is written as and this operation can be easily implemented in MATLAB as shown below. clear all close all fm=100; fs=1000; Ts=1/fs; N=1024; t=0:Ts:(N-1)*Ts; x=cos(2*pi*fm*t); W=exp(-j*2*pi/N); n=0:N-1; for k=0:N-1 X(k+1)=sum(x.*(W.^(k*n))); end plot((0:k)/N,abs(X)) xlabel (‘Normalized Frequency’) ylabel (‘X’) The DFT of the cosine […]