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Graduate Course in Radio Astronomy

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Fourier Analysis

Nicole Kaiser, December 12, 2005

Fourier Series and Transforms

Fourier analysis is used to examine the spectrum of time-sampled data. The incoming signal to the spectrometer can be represented by a Fourier series (eq. 1), which defines a periodic function as an infinite sum of sines and cosines:

sines and cosines (1)

or in complex form, where e=cos(θ) + isin(θ):

complex form (2)

In order to examine the frequency components of the Fourier series, a Fourier transform must be applied. This transforms the data from the time domain into the frequency domain. The fourier transform of the time window is the point spread function in the frequency domain. Equations (3) and (4) are Fourier transform pairs, where f(t) is the time-sampled data, and F(ω) is the frequency spectrum. F(ω) is generally referred to as the forward Fourier transform, and f(t) as the inverse Fourier transform:

Forward Fourier Transform (3)

Inverse Fourier Transform (4)

The above equations are used for continuously sampled data. For more practical purposes, data is usually sampled in discrete time intervals, rather than continuously. The discrete Fourier transform (DFT) is then used to obtain a frequency spectrum of such data:

Discrete Fourier Transform (5)

Discrete Fourier Transform (6)

The discrete Fourier transform is highly symmetric, allowing for a much faster method of transforming the data, known as the fast Fourier transform (FFT). The FFT is a discrete Fourier transform algorithm which speeds up computation time by reducing the number of computations. When sampling N points in the time domain, a regular Fourier transform requires N2 operations (where an operation is a complex multiplication followed by a complex addition)1. The FFT is able to compute the result in less than log2N operations. The base-two dependence requires the number of sampling points in the time-domain to be twice the number of points in the frequency domain.

Data Sampling

Sampling Rate

Two important factors which must be considered when performing a Fourier transform are the sampling rate and the size and shape of the sampling window. These factors will have an effect on the resolution and quality of the resulting spectrum.

The rate at which the data is sampled defines how many points will be transformed from the time domain to the frequency domain. This rate can be optimized to ensure the best quality of data. For example, if you are examining a particular bandwidth in the frequency domain, you need to ensure you are properly sampling the waveform in order to properly construct its spectrum.

Sampling

Figure 1: (a)Analog-to-digital conversion of a sinusoid of a certain frequency. (b) At higher input frequencies, related in a special way to the original input frequency, exactly the same digitized output occurs. (c) Shows how the A/D conversion of the two sinusoids can produce the same result. (Karl, 1989)

To ensure the waveform is properly represented in the frequency domain, it must be sampled at twice its frequency. Thus, your sampling rate must be at least twice the highest frequency in your bandwidth. The highest frequency that can be reconstructed accurately is known as the Nyquist frequency, and it is equal to half the sampling rate.

Sampling Window

The amount of resolution in the frequency domain is dependent upon the size and shape of the sampling window. The longer the length of the window in the time domain, the higher the resolution in the frequency domain. The most basic sampling window shape is the boxcar (rectangular function) (figure 2a).

(Fig 1 - boxcar and sinc

Figure 2: (a) Boxcar Function (b) Sinc Function - the Fourier transform of the boxcar (Karl, 1989)

It looks like an ideal window shape in the time domain, however, the Fourier transform of the boxcar is a sinc function (figure 2b). The only advantage of the sinc function is that it has a relatively narrow central peak, and thus it has good spectral resolution. However, the large side lobes of the sinc function greatly decrease the quality of the frequency spectrum. The Bartlett (triangle) function (figure 3) is used to slowly bring the edges of the window down to zero. When transformed to the frequency spectrum, the side lobes are greatly reduced, but as result the central peak becomes wider, causing a decrease in spectral resolution. A more ideal sampling window in both the time and frequency domain is a weighted function, such as the Hanning window (figure 3). This window is tapered to reduce side lobes in the spectrum, but does not fall to zero as slowly as the Bartlett function, and so the frequency resolution is slightly improved. The size and shape of the sampling window are very important in obtaining the desired results, and usually the best window involves sacrificing resolution for improved data quality.

Hanning Windows

Figure 3: commonly used windows (Karl, 1989)
  1. Cooley, J.W. and Tukey, O.W. "An Algorithm for the Machine Calculation of Complex Fourier Series". Math. Comput. 19, 297-301, 1965.
  2. Karl, J.H. An Introduction to Digital Signal Processing. Academic Press, Inc., USA, 1989.
  3. Mathworld http://mathworld.wolfram.com/FourierTransform.html

Last modified: 10:22 am July 17, 2014

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