Digital Signal Processing: A Practitioner's Approach

4

Fourier Transform and Signal Spectrum

Any book on DSP is not complete without a detailed discussion of the Fourier transform. There are many publications in this area, but Brigham [1] is probably the best. The fast Fourier transform (FFT) is the numerical method [2] for computing the discrete Fourier transform (DFT) of a given time series. It exploits the periodicity properties of the function e^j2πk where k is an integer.

4.1 Heterodyne Spectrum Analyser

Analysing the spectrum using the superheterodyning principle is very popular due to the simple nature of its implementation. The idea is to create a relative motion between the bandpass filter (BPF) and the incoming signal spectrum:

Equation (4.1) is shown in Figure 4.1 to give a better insight. The digitised signal s_k is multiplied with a frequency-swept signal x_k. Note that we need to sweep using a staircase function¹ not a ramp function. We have presented the results at SNR = 0 dB.

4.1.1 Spectrum Analyser

The signal x_k is defined as

Figure 4.1 An engineer's view of (4.1)

Figure 4.2 Digital spectrum analyser

The frequency f_k is ...