Q2.4 > T. Determine the Fourier transformof g(t). What happens to this transform when we allow t to be-come zero? Hint: Express g(t) as the superposition of two signals,one corresponding to...

Q2.4

Extracted text: T) algorithms were brought into prominence by the publication of the - (1965). For discussions of FFT algorithms, see the books by Oppen- 1999, Ch. 9) and Haykin and Van Veen (2005). For a discussion of y be used to perform linear filtering, see the book Numerical Recipes 02). (a) Evaluate the even and odd parts of a rectangular pulse de- fined by g(t) = A rect 2 (b) What are the Fourier transforms of these two parts of the pulse? |G(H|| arg |G(f)|| 1.0 2 -W W -W W T 2 Figure P2.4 2.4 Determine the inverse Fourier transform of the frequency function G(f) defined by the amplitude and phase spectra shown in Figure P2.4. 2.5 The following expression may be viewed as an approxi- mate representation of a pulse with finite rise time: pt + T Tu? du 1 g(t): exp where it is assumed that T >> T. Determine the Fourier transform of g(t). What happens to this transform when we allow t to be- come zero? Hint: Express g(t) as the superposition of two signals, one corresponding to integration from t – T to 0, and the other from 0 to t + T. 2.6 The Fourier transform of a signal g(t) is denoted by G(f). Prove the following properties of the Fourier transform: (a) If a real signal g(t) is an even function of time t, the Fourier transform G(f) is purely real. If a real signal g(t) is an odd function of time t the Fourier transform G(f) is purely