Signals and Systems - Electrical Engineering

424 CHAPTER 7: Sampling Theory (a) (b) (c) 1 −Ωmax Ωmax X(Ω) Ω No aliasing Ωmax Ωs≥ 2 Ωmax Xs(Ω) 1/Ts Ωs Ω · · · · · · Aliasing ...

7.2 Uniform Sampling 425 wheremaxis the maximum frequency inx(t)—can be sampled uniformly and without frequency aliasing using ...

426 CHAPTER 7: Sampling Theory 0 1 2 3 − 2 − 1 0 1 2 x( t), x(0.2 n) (a) 0 1 2 3 − 2 − 1 0 1 2 x( t), x(0.4 n) (b) 0 1 2 3 − 2 − ...

7.2 Uniform Sampling 427 nExample 7.2 Consider the following signals: (a) x 1 (t)=u(t+0.5)−u(t−0.5) (b) x 2 (t)=e−tu(t) Determin ...

428 CHAPTER 7: Sampling Theory We found that forM= 20 πrad/sec 98.9% of the energy of the signal is included, and thus it could ...

7.2 Uniform Sampling 429 Bandlimited or Not? The following, taken from David Slepian’s paper “On Bandwidth” [66], clearly descri ...

430 CHAPTER 7: Sampling Theory FIGURE 7.4 Anti-aliasing filtering of non-band-limited signal. −Ωc −Ωc Ωc Ωc Ω 1 X(Ω) Xa(Ω) H(s) ...

7.2 Uniform Sampling 431 Show that if we sample these signals usingTs= 2 π/s, we cannot differentiate the sampled signals (i.e. ...

432 CHAPTER 7: Sampling Theory FIGURE 7.5 Sampling of two sinusoids of frequencies 0 = 1 and  0 +s= 8 withTs= 2 π/s. The hig ...

7.2 Uniform Sampling 433 = ∫∞ −∞ [ ∑ n x(nTs)δ(τ−nTs) ] hlp(t−τ)dτ = ∑ n x(nTs) sin(π(t−nTs)/Ts) π(t−nTs)/Ts (7.16) after replac ...

434 CHAPTER 7: Sampling Theory 0 1 2 3 4 5 67 − 1 0 1 t (sec) x( t) , y (t) x( t) x (r t) (a) (b) − 10 − 8 − 6 − 4 − 2 0 2 4 6 8 ...

7.2 Uniform Sampling 435 0 1 2 3 4 5 6 78 − 1 0 1 t (sec) 0 0.5 1 1.5 2 2.5 3 3.5 −1.5 − 1 −0.5 0 0.5 1 1.5 0 0.0050.010.0150.02 ...

436 CHAPTER 7: Sampling Theory 0 2 4 6 0 1 2 3 4 x( t), y( t) − 10 − 5 0 5 0 0.5 1 1.5 2 f (KHz) −0.4−0.2 0 0.2 0 0.5 1 1.5 2 f ...

7.3 The Nyquist-Shannon Sampling Theorem 437 xr = xr+x(k)∗sinc(t−(k−1)); end xx(1:100:N∗100) = x(1:N); xx = [xx zeros(1,99)]; NN ...

438 CHAPTER 7: Sampling Theory Origins of the Sampling Theory — Part 2 As mentioned in Chapter 0, the theoretical foundations of ...

7.4 Practical Aspects of Sampling 439 nExample 7.4 Consider the development of an AM transmitter that uses a computer to generat ...

440 CHAPTER 7: Sampling Theory FIGURE 7.10 Sampling using a sample-and-hold system(δ=Ts). Δ t 1 × t t 0 x(t) δTs(t) xs(t) h(t) y ...

7.4 Practical Aspects of Sampling 441 FIGURE 7.11 Sample-and-hold circuit. r C R ++ − − Ts x(t) xsh(t) n Two significant changes ...

442 CHAPTER 7: Sampling Theory FIGURE 7.12 Four-level quantizer and coder. Δ Δ 10 11 01 00 − 2 Δ −Δ x(nTs) −Δ − 2 Δ 2 Δ x∧(nTs) ...

7.4 Practical Aspects of Sampling 443 and use the characterization of the quantizer given in Equation (7.24), we have then that ...