Signals and Systems - Electrical Engineering

304 CHAPTER 5: Frequency Analysis: The Fourier Transform (c) The Laplace transform ofx 3 (t)is X 3 (s)= 1 s+ 1 + 1 −s+ 1 = 2 1 − ...

5.5 Linearity, Inverse Proportionality, and Duality 305 way. This is so even though the time signals add correctly tox(t). The F ...

306 CHAPTER 5: Frequency Analysis: The Fourier Transform or a sinc function where Aτ corresponds to the area under x 3 (t). The ...

5.5 Linearity, Inverse Proportionality, and Duality 307 − 3 − 2 − 1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 t − 3 − 2 − 1 0 1 2 3 0 0.2 0.4 ...

308 CHAPTER 5: Frequency Analysis: The Fourier Transform This property is shown by a change of variable in the integration, F[x( ...

5.5 Linearity, Inverse Proportionality, and Duality 309 FIGURE 5.2 (a) Pulsex(t)and its compressed versionx 1 (t)=x( 2 t), and ( ...

310 CHAPTER 5: Frequency Analysis: The Fourier Transform FIGURE 5.3 Magnitude and phase spectrum of two-sided signalx(t)=e−|t|. ...

5.5 Linearity, Inverse Proportionality, and Duality 311 This can be shown by considering the inverse Fourier transform x(t)= 1 2 ...

312 CHAPTER 5: Frequency Analysis: The Fourier Transform − 1 −0.5 0 0.5 1 0 2 4 6 8 10 t − 100 − 50 0 50 100 0 5 10 Ω − 1 −0.5 0 ...

5.6 Spectral Representation 313 where we used the Laplace transform ofδ(t−ρ 0 )+δ(t+ρ 0 ), which ise−sρ^0 +esρ^0 defined over th ...

314 CHAPTER 5: Frequency Analysis: The Fourier Transform Applying the frequency shifting to x(t)cos( 0 t)=0.5x(t)ej^0 t+0.5x(t ...

5.6 Spectral Representation 315 Solution The modulated signals are y 1 (t)=x 1 (t)cos( 10 t)=e−|t|cos( 10 t) −∞<t<∞ y 2 ( ...

316 CHAPTER 5: Frequency Analysis: The Fourier Transform (a) (b) − 5 05 −0.5 0 0.5 1 t(sec) y^1 (t ) − 20 0 20 0 0.2 0.4 0.6 0.8 ...

5.6 Spectral Representation 317 signals. When radiating a signal with an antenna, the length of the antenna is about a quarter o ...

318 CHAPTER 5: Frequency Analysis: The Fourier Transform Remarks n When plotting|X()|versus, which we call the Fourier magnitu ...

5.6 Spectral Representation 319 so that the Fourier coefficients ofx(t)are (T 0 =1): Xk= 1 T 0 X 1 (s)|s=j 2 πk= 1 (j 2 πk)^2 2 ...

320 CHAPTER 5: Frequency Analysis: The Fourier Transform %%%%%%%%%%%%%%%%%%%%% % Example 5.8---Fourier series %%%%%%%%%%%%%%%%%% ...

5.6 Spectral Representation 321 This energy conservation property is shown using the inverse Fourier transform. The finite-energ ...

322 CHAPTER 5: Frequency Analysis: The Fourier Transform Solution The energy of the pulse isE=2 (the area under the pulse). But ...

5.6 Spectral Representation 323 since the integral can be thought of as an infinite sum of complex values. Comparing the two int ...