Fourier Transforms in Radar And Signal Processing_4

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Fourier Transforms in Radar And Signal Processing_4 Pulse Spectra 53spectral power factors (in both linear and logarithmic form) multiplying theoriginal pulse spectrum in the two cases, sinc2 2f for the rectangular pulseand 1/[1 + (2 f )2 ] for the stray capacitance. The power spectrum of thesmoothed pulse is that of the spectrum of the original pulse multiplied byone of these spectra. Assuming the smoothing impulse response is fairlyshort compared with the pulse length, the spectrum of the pulse will bemainly within the main lobe of the impulse response spectrum. We see thatthe side-lobe pattern of the pulse will be considerably reduced by the smooth-ing (e.g., by about 10 dB at ± 0.4/ from center frequency). We also seethat the rect pulse of width 2 gives a response fairly close to the straycapacitance filter with time constant .3.7 General Rounded Trapezoidal PulseHere we consider the problem of rounding the four corners of a trapezoidalpulse over different time intervals. This may not be a particularly likelyproblem to arise in practice in connection with radar, but the solution tothis awkward case is interesting and illuminating, and could be of use insome other application. The problem of the asymmetrical trapezoidal pulse was solved in Section3.4 by forming the pulse from the difference of two step-functions, each ofwhich was convolved with a rectangular pulse to form a rising edge. Byusing different-width rectangular pulses, we were able to obtain differentslopes for the front and back edges of the pulse. In this case we extend this principle by expressing the convolving rectpulses themselves as the difference of two step functions. The (finite) risingedge can then be seen to be the difference of two infinite rising edges, asshown in Figure 3.14. Each of these, which we call Ramp functions, isproduced by the convolution of two unit step functions as shown in Figure3.15 and defined in (3.20) below. We define the Ramp function, illustrated in Figure 3.15, by Ramp (t − T ) = h (t ) ⊗ h (t − T ) (3.20)so that for t ≤ 0 0 Ramp (t ) = (t ∈ ) (3.21) for t > 0 t 54 Fourier Transforms in Radar and Signal ProcessingFigure 3.14 Rising edge as the difference of two Ramp functions. Pulse Spectra 55Figure 3.15 Ramp function.(A different, finite, linear function is required in Chapter 6; this is calledramp.) Having now separated the four corners of the trapezoidal pulse intothe corners of four Ramp functions, they can now all be rounded separatelyby convolving the Ramp functions with different-width rect functions (orother rounding functions, if required) as in Figure 3.11, before combiningto form the smoothed pulse. Before obtaining the Fourier transform of therounded pulse, we obtain the transform of the trapezoidal pulse in the formof the four Ramp functions (two for each of the rising and falling edges). In mathematical notation, the rising edge of Figure 3.14 can beexpressed in the two ways t − T0 h (t ) ⊗ rect = h (t ) ⊗ (Ramp (t − T1 ) − Ramp (t − T2 )) T (3.22) The Fourier transform of the left side is, from P2a, P3a, R7b, R5, andR6a, (f ) 1 + T sinc f T exp (−2 if T0 ) (3.23) 2 2 if ( f ) sinc f T exp (−2 if T0 ) = + T 2 2 ifwhere we have used ( f − f 0 ) u ( f ) = ( f ) u ( f 0 ) in general, so ( f ) sinc ( f T ) = ( f ). The transform of the difference of the Rampfunctions on the right side is, using (3.20), P2a, R7b, and R6a, (f ) 1 (f ) 1 + + [exp (−2 if T1 ) − exp (−2 if T2 )] 2 2 if 2 2 if (3.24)56 Fourier Transforms in Radar and Signal Processing Using T 0 and T as given in Figure 3.14, the difference of the expo-nential terms becomes exp (−2 if T0 ) (exp (2 if T ) − exp (−2 if T )) or2i sin (2 f T ) exp (−2 if T0 ), so again using ( f − f 0 ) u ( f ) = ( f ) u ( f 0 )[with u ( f 0 ) = ...