Signals and Systems - Electrical Engineering

264 C H A P T E R 4: Frequency Analysis: The Fourier Series Fork6=0 we have|Yk|=|Xk|/(πk), so that askincreases the frequency co ...

4.7 Convergence of the Fourier Series 265 (a) (b) 0 0.5 1 1.5 2 − 1 −0.5 0 0.5 1 t Period x( t) (^00204060) 0.1 0.2 0.3 0.4 Ω |X ...

266 C H A P T E R 4: Frequency Analysis: The Fourier Series conditions under which the series converges, we need to classify sig ...

4.7 Convergence of the Fourier Series 267 with respect to the Fourier coefficientsXk. To minimizeENwith respect to the coefficie ...

268 C H A P T E R 4: Frequency Analysis: The Fourier Series FIGURE 4.12 Approximate Fourier seriesxN(t)of the pulse trainx(t)(di ...

4.7 Convergence of the Fourier Series 269 nExample 4.10 Consider the mean-square error optimization to obtain an approximation o ...

270 C H A P T E R 4: Frequency Analysis: The Fourier Series nExample 4.11 Consider the train of pulses in Example 4.5. Determine ...

4.8 Time and Frequency Shifting 271 we then have that x(t−t 0 )= ∑ k Xkejk^0 (t−t^0 )= ∑ k [ Xke−jk^0 t^0 ] ejk^0 t x(t+t 0 ) ...

272 C H A P T E R 4: Frequency Analysis: The Fourier Series nExample 4.12 To illustrate the modulation property using MATLAB con ...

4.9 Response of LTI Systems to Periodic Signals 273 FIGURE 4.13 (a) Modulated square-wave x 1 (t)cos( 20 πt)and (b) cosine x 2 ( ...

274 C H A P T E R 4: Frequency Analysis: The Fourier Series If we callYk=XkH(jk 0 )we have a Fourier series representation ofys ...

4.9 Response of LTI Systems to Periodic Signals 275 M = length(x); figure(1) x1 = [zeros(1, 5) x(1:M)]; z = y(1); y1 = [zeros(1, ...

276 C H A P T E R 4: Frequency Analysis: The Fourier Series If the inputx(t)of a causal and stable LTI system, with impulse resp ...

4.9 Response of LTI Systems to Periodic Signals 277 nExample 4.13 To illustrate the filtering of a periodic signal, consider a z ...

278 C H A P T E R 4: Frequency Analysis: The Fourier Series Because the magnitude response of the low-pass filter changes very l ...

4.10 Other Properties of the Fourier Series 279 4.10 Other Properties of the Fourier Series In this section we present additiona ...

280 C H A P T E R 4: Frequency Analysis: The Fourier Series TheXkare purely imaginary. Indeed, for an oddx(t), Xk= 1 T 0 ∫ T 0 x ...

4.10 Other Properties of the Fourier Series 281 FIGURE 4.16 Nonsymmetric periodic signals. 2 − 2 − 1 0 1 − 2 − 1 01 2 3 2 (^23) ...

282 C H A P T E R 4: Frequency Analysis: The Fourier Series FIGURE 4.17 Even and odd components of the period ofy(t), − 1 ≤t≤ 1. ...

4.10 Other Properties of the Fourier Series 283 T 0 =MT 2 =NT 1 , and its Fourier coefficients are Zk=αXk/N+βYk/M for integersks ...