Signals and Systems - Electrical Engineering

324 CHAPTER 5: Frequency Analysis: The Fourier Transform function by lettings=j, (2) by using the symbolic functionfourier, and ...

5.6 Spectral Representation 325 The Fourier transform ofx 2 (t)=e−tu(t)is X 2 ()= 1 1 +j The magnitude and phase are given by ...

326 CHAPTER 5: Frequency Analysis: The Fourier Transform 0 2 4 −0.2 0 0.2 0.4 0.6 0.8 1 t(sec) x^1 (t ) − 20 − 2 020 0 0.2 0.4 0 ...

5.7 Convolution and Filtering 327 This is also a low-pass signal likex 2 (t)in Example 5.11, but this is “smoother” than that on ...

328 CHAPTER 5: Frequency Analysis: The Fourier Transform This can be shown by considering the eigenfunction property of LTI syst ...

5.7 Convolution and Filtering 329 or H(j)= Y() X() (5.20) The magnitude and the phase ofH(j)are the magnitude and phase freq ...

330 CHAPTER 5: Frequency Analysis: The Fourier Transform the desirable frequency band or bands, and let it be close to zero in t ...

5.7 Convolution and Filtering 331 so thatAX 0 =1, orA= 1 /X 0 =π/2, to get the output to have a unit amplitude. Although the pro ...

332 CHAPTER 5: Frequency Analysis: The Fourier Transform multiplied by a constant. This is one more example of the inverse relat ...

5.7 Convolution and Filtering 333 FIGURE 5.8 Ideal filters: (a) low pass, (b) band pass, (c) band eliminating, and (d) high pass ...

334 CHAPTER 5: Frequency Analysis: The Fourier Transform and stable filter with frequency response H(j)should satisfy the condi ...

5.7 Convolution and Filtering 335 the output of the filter is the signal xN(t)=F−^1 [X()H(j)] =F−^1   ∑N k=−N 2 πXkδ(−k 0 ...

336 CHAPTER 5: Frequency Analysis: The Fourier Transform the capacitor is an open circuit (its impedance would be infinite), so ...

5.7 Convolution and Filtering 337 5.7.3 Frequency Response from Poles and Zeros.................................. Given a ration ...

338 CHAPTER 5: Frequency Analysis: The Fourier Transform So that for 0≤ 0 <∞, if we compute the length and the angle ofZE( ...

5.7 Convolution and Filtering 339 Since there are no zeros, the frequency response of this filter depends inversely on the behav ...

340 CHAPTER 5: Frequency Analysis: The Fourier Transform n Zeros create “valleys” at the frequencies in the jaxis in front of t ...

5.7 Convolution and Filtering 341 wmax = 10; % maximum frequency [w, Hm, Ha] = freqresps(n, d, wmax); % frequency response splan ...

342 CHAPTER 5: Frequency Analysis: The Fourier Transform 0510 0 0.2 0.4 0.6 0.8 1 |H (j Ω )| Ω Magnitude Response 0510 − 50 0 50 ...

5.7 Convolution and Filtering 343 0 0.2 0.4 0.6 0.8 1 |H (j ω )| Ω 0510 Magnitude Response 0510 Ω − 100 0 100 ∠ H (j ω) Phase Re ...