Caner Ozdemir - Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms

Figure 1.6 An ideal and real LP filter characteristics.

Windowing procedure is usually applied to smoothen a time‐domain signal, therefore, filtering out higher frequency components. Some of the popular windows that are widely used in signal and image processing are Kaiser, Hanning, Hamming, Blackman, and Gaussian. A comparative plot of some of these windows is given in Figure 1.7.

The effect of windowing operation is illustrated in Figure 1.8. A time‐domain signal of a rectangular signal is shown in Figure 1.8a and its FT is provided in Figure 1.8b. This function is, in fact, a sinc ( sinus cardinalis ) function and has major side lobes. For the sinc function, the highest side lobe is approximately 13 dB lower than the apex of the main lobe. This much of contrast, of course, may not be sufficient in some imaging applications. As shown in Figure 1.8c, the original rectangular time‐domain signal is Hanning windowed. Its corresponding spectrum is depicted in Figure 1.8d where the side lobes are highly suppressed thanks to the windowing operation. For this example, the highest side lobe level is now 32 dB below the maximum value of the main lobe which provides better contrast when compared to the original, non‐windowed signal.

Figure 1.7 Some common window characteristics.

Figure 1.8 Effect of windowing. (a) Rectangular time signal, (b) its Fourier spectrum: a sinc signal, (c) Hanning windowed time signal, (d) corresponding frequency‐domain signal.

A main drawback of windowing is the resolution decline in the frequency signal. The FT of the windowed signal has worse resolution than the FT of the original time‐domain signal. This feature can also be noticed from the example in Figure 1.8. By comparing the main lobes of the figures on the right, the resolution after windowing is almost twice as bad when compared to the original frequency‐domain signal. A comprehensive examination of windowing procedure will be presented later on, in Chapter 5.

Sampling can be regarded as the preprocess of transforming a continuous or analog signal to a discrete or digital signal. When the signal analysis has to be done using digital computers via numerical evaluations, continuous signals need to be converted to the digital versions. This is achieved by applying the common procedure of sampling. Analog‐to‐digital (A/D) converters are common electronic devices to accomplish this process. The implementation of a typical sampling process is shown in Figure 1.9. A time signal s ( t ) is sampled at every T sseconds such that the discrete signal, s [ n ], is generated via the following equation:

Figure 1.9 Sampling. (a) continuous time signal, (b) discrete‐time signal after the sampling.

(1.20)

Therefore, the sampling frequency f sis equal to 1/ T swhere T sis called the sampling interval .

A sampled signal can also be regarded as the digitized version of the multiplication of the continuous signal, s ( t ) with the impulse comb waveform, c ( t ) as depicted in Figure 1.10.

According to the Nyquist–Shannon sampling theorem, the perfect reconstruction of the signal is only possible provided that the sampling frequency f sis equal or larger than twice the maximum frequency content of the sampled signal (Shannon 1949). Otherwise, signal aliasing is unavoidable and only distorted version of the original signal can be reconstructed.

Figure 1.10 Impulse comb waveform composed of ideal impulses.

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 owhere N is the number of samples in one period. Then, the Fourier integral in Eq. 1.1will turn to a summation as shown below.

(1.21)

Dropping the f oand T sinside 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 oand 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 oand T sinside the parenthesis, the IDFT of a discrete frequency signal S [ k ] is given as

(1.24)

FFT is the efficient and fast way of evaluating the DFT of a signal. Normally, computing the DFT is in the order of N 2arithmetic 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 1.11where 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.