Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms. Caner Ozdemir

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discrete‐time signal after the sampling.

      (1.20)equation

      Therefore, the sampling frequency fs is equal to 1/Ts where Ts is called the sampling interval.

Graph depicts the impulse comb waveform composed of ideal impulses.

      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 To = 1/fo. Then, its sampled (or discrete) version is s[n] ≜ s(nTs) with a period of NTs = To 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)equation

      Dropping the fo and Ts 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)equation

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

      (1.23)equation

      (1.24)equation

      1.7.2 FFT

      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 To = 1/fo sampled N times with a sampling interval of Ts = To/N, the frequency resolution (or the sampling interval in frequency) after applying the DFT can be found as

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

Graphs depict the example of DFT operation: (a) discrete time-domain signal, (b) discrete frequency-domain signal without FFT shifting, (c) discrete frequency-domain signal with FFT shifting.

      Similar arguments can be made for the case of IDFT. Considering a discrete frequency‐domain signal with a bandwidth of B sampled N times with a sampling interval of Δf, the time resolution (or the sampling interval in time) after applying IDFT can be found as

      The

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