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Figure 1.10 Impulse comb waveform composed of ideal impulses.

1.7 DFT and FFT

      1.7.1 DFT

      As explained in Section 1.1, the FT is used to transform continuous signals from one domain to another. It is usually used to describe the continuous spectrum of an aperiodic time signal. To be able to utilize the FT while working with digital signals, the digital or DFT has to be used.

      Let s(t) be a continuous periodic time signal with a period of T_o = 1/f_o. Then, its sampled (or discrete) version is s[n] ≜ s(nT_s) with a period of NT_s = T_o where N is the number of samples in one period. Then, the Fourier integral in Eq. 1.1 will turn to a summation as shown below.

      (1.21)

      Dropping the f_o and T_s inside the parenthesis for the simplicity of nomenclature and therefore switching to discrete notation, DFT of the discrete signal s[n] can be written as

      (1.22)

      In a dual manner, let S(f) represent a continuous periodic frequency signal with a period of Nf_o = N/T_o and let S[k] ≜ S(kf_o) be the sampled signal with the period of Nf_o = f_s. Then, the IDFT of the frequency signal S[k] is given by

      (1.23)

Using the discrete notation by dropping the f_o and T_s inside the parenthesis, the IDFT of a discrete frequency signal S[k] is given as

      (1.24)

      1.7.2 FFT

FFT is the efficient and fast way of evaluating the DFT of a signal. Normally, computing the DFT is in the order of N² arithmetic operations. On the other hand, fast algorithms like Cooley‐Tukey's FFT technique produce arithmetic operations in the order of N log(N) (Cooley and Tukey 1965; Brenner and Rader 1976; Duhamel 1990). An example of DFT is given is Figure 1.11 where a discrete time‐domain ramp signal is plotted in Figure 1.11a and its frequency‐domain signal obtained by an FFT algorithm is given in Figure 1.11b.

      1.7.3 Bandwidth and Resolutions

      The duration, the bandwidth, and the resolution are important parameters while transforming signals from time domain to frequency domain or vice versa. Considering a discrete time‐domain signal with a duration of T_o = 1/f_o sampled N times with a sampling interval of T_s = T_o/N, the frequency resolution (or the sampling interval in frequency) after applying the DFT can be found as

(1.25)

      The spectral extend (or the frequency bandwidth) of the discrete frequency signal is

(1.26)

For the example in Figure 1.11, the signal duration is 1 ms with N = 10 samples. Therefore, the sampling interval is 0.1 ms. After applying the expressions in Eqs. 1.25 and 1.26, the frequency resolution is 100 Hz and the frequency bandwidth is 1000 Hz. After taking the DFT of the discrete time‐domain signal, the first entry of the discrete frequency signal corresponds to zero frequency and negative frequencies are located in the second half of the discrete frequency signal as seen in Figure 1.11b. After the DFT operation, therefore, the entries of the discrete frequency signal should be swapped from the middle to be able to form the frequency axis correctly as shown in Figure 1.11c. This property of DFT will be thoroughly explored in Chapter 5 to demonstrate its use in ISAR imaging.