Signals and Systems - Electrical Engineering

244 C H A P T E R 4: Frequency Analysis: The Fourier Series circuit shown in Figure 4.1 where the input is vs(t)= 1 +cos(10,000t ...

4.3 Complex Exponential Fourier Series 245 since H(j 0 )= 1 H(j10,000)≈ 1 j 104 = −j 10,000 H(−j10,000)≈ 1 −j 104 = j 10,000 Thu ...

246 C H A P T E R 4: Frequency Analysis: The Fourier Series perpendicularity of vectors: Perpendicular vectors cannot be represe ...

4.3 Complex Exponential Fourier Series 247 where the Fourier coefficientsXkare found according to Xk= 1 T 0 t (^0) ∫+T 0 t 0 x(t ...

248 C H A P T E R 4: Frequency Analysis: The Fourier Series n The Fourier coefficients{Xk}are easily obtained using the orthonor ...

4.4 Line Spectra 249 The power of a periodic signalx(t)of periodT 0 is given by Px= 1 T 0 ∫ T 0 |x(t)|^2 dt Replacing the Fourie ...

250 C H A P T E R 4: Frequency Analysis: The Fourier Series The power line spectrum|Xk|^2 versusk 0 ofx(t)displays the distribu ...

4.5 Trigonometric Fourier Series 251 4.5 Trigonometric Fourier Series Thetrigonometric Fourier seriesof a real-valued, periodic ...

252 C H A P T E R 4: Frequency Analysis: The Fourier Series Let us then show how the coefficientsckanddkcan be obtained directly ...

4.5 Trigonometric Fourier Series 253 Finally, since the exponential basis{ejk^0 t}={cos(k 0 t)+jsin(k 0 t)}, the sinusoidal b ...

254 C H A P T E R 4: Frequency Analysis: The Fourier Series FIGURE 4.2 (a) Magnitude (top left) and phase (bottom left) line spe ...

4.6 Fourier Coefficients from Laplace 255 0 0.2 0.4 0.6 0.8 1 Magnitude line spectrum |Y |k − 200 0 200 − 1 0 1 Phase line spect ...

256 C H A P T E R 4: Frequency Analysis: The Fourier Series For a periodic signalx(t), of periodT 0 , if we know or can easily c ...

4.6 Fourier Coefficients from Laplace 257 the Fourier coefficients will be real. Doing this analysis before the computations is ...

258 C H A P T E R 4: Frequency Analysis: The Fourier Series %%%%%%%%%%%%%%%%% % Example 4.5---Fourier series of train of pulses ...

4.6 Fourier Coefficients from Laplace 259 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 Period x( t) t 0 50 100 0 0.2 0.4 0.6 0.8 1 Magnitud ...

260 C H A P T E R 4: Frequency Analysis: The Fourier Series Notice that about 11 of them (including the zero values), or the dc ...

4.6 Fourier Coefficients from Laplace 261 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t Period 0 50 100 0 0.1 0.2 0.3 0.4 0.5 0.6 Ma ...

262 C H A P T E R 4: Frequency Analysis: The Fourier Series since cos(πk)=(− 1 )k. The DC value of the full-wave rectified signa ...

4.6 Fourier Coefficients from Laplace 263 so that the Fourier coefficients are given by(T 0 =2, 0 =π): Yk= 1 T 0 Y 1 (s)|s=jok ...