ECE-2025 Fall-2004 Lecture 22: Sampling and Reconstruction (Fourier View), Lab Reports of Electrical and Electronics Engineering

Download ECE-2025 Fall-2004 Lecture 22: Sampling and Reconstruction (Fourier View) and more Lab Reports Electrical and Electronics Engineering in PDF only on Docsity! ECE-2025 Fall-2004 Lecture 22 Sampling and Reconstruction (Fourier View) 12-Nov-04 11/10/2004 ECE-2025 Fall-03 jMc 2 Info: Web-CT, Lab, HW Lab #11 AM system for transmission Single Sideband for scrambling speech CHECK YOUR GRADES !!! Web-CT is the OFFICIAL gradebook Quiz #3 will be 19-Nov (Friday) Coverage: HW #9, 10, 11 and 12 Chapters 7, 9, 10, and 11 Review Session, 18-Nov, Thurs @ 7:30pm 11/10/2004 ECE-2025 Fall-03 jMc 3 Pop Quiz ( ) ( ) ?)1()( ?)1()( ?)1()( 2 2 2 =− =+ =+∗ ∫ ∞ ∞− − − − dtttue ttue ttue t t t δ δ δ 11/10/2004 ECE-2025 Fall-03 jMc 4 Lectures A Lecture is the process in which the notes of the professor become the notes of the students … without passing through the minds of either. Lecture 11/10/2004 ECE-2025 Fall-03 jMc 5 LECTURE OBJECTIVES Sampling Theorem Revisited GENERAL: in the FREQUENCY DOMAIN Fourier transform of sampled signal Reconstruction from samples Reading: Chap 12, Section 12-3 Review of FT properties Convolution multiplication Frequency shifting Review of AM 11/10/2004 ECE-2025 Fall-03 jMc 6 Table of FT Properties x(t − td ) ⇔ e − jωtd X( jω ) x(t)e jω0t ⇔ X( j(ω − ω0)) Delay Property Frequency Shifting x(at) ⇔ 1|a | X( j(ωa )) Scaling x(t) ∗h(t) ⇔ H( jω )X( jω ) 11/10/2004 ECE-2025 Fall-03 jMc 7 Amplitude Modulator x(t) modulates the amplitude of the cosine wave. The result in the frequency-domain is two SHIFTED copies of X(jω). y(t) = x(t)cos(ωct +ϕ ) X( jω ) x(t) cos(ωct + ϕ) Y ( jω) = 12 e jϕX( j(ω −ωc)) + 12 e− jϕX( j(ω + ωc)) Phase 11/10/2004 ECE-2025 Fall-03 jMc 8 DSBAM: Frequency-Domain “Typical” bandlimited input signal Frequency-shifted copies ))((2 1 c j jXe ωωϕ +− ))((21 c j jXe ωωϕ − Upper sidebandLower sideband )( ωjX 11/10/2004 ECE-2025 Fall-03 jMc 17 Illustration of Sampling x(t) x[n] = x(nTs ) ∑ ∞ −∞= −= n sss nTtnTxtx )()()( δ n t 11/10/2004 ECE-2025 Fall-03 jMc 18 Sampling: Freq. Domain EXPECT FREQUENCY SHIFTING !!! ∑∑ ∞ −∞= ∞ −∞= =−= k tjk k n s seanTttp ωδ )()( ∑ ∞ −∞= = k tjk k sea ω 11/10/2004 ECE-2025 Fall-03 jMc 19 Frequency-Domain Analysis xs (t) = x(t) δ (t − nTs ) n=−∞ ∞ ∑ = x(nTs )δ (t − nTs ) n=−∞ ∞ ∑ xs (t) = x(t) 1 Tsk=−∞ ∞ ∑ e jkωst = 1Ts x(t) k=−∞ ∞ ∑ e jkωst Xs ( jω) = 1 Ts X( j(ω k=−∞ ∞ ∑ − kωs )) ωs = 2π Ts 11/10/2004 ECE-2025 Fall-03 jMc 20 Frequency-Domain Representation of Sampling Xs ( jω) = 1 Ts X( j(ω k=−∞ ∞ ∑ − kωs )) “Typical” bandlimited signal 11/10/2004 ECE-2025 Fall-03 jMc 21 Aliasing Distortion If ωs < 2ωb , the copies of X(jω) overlap, and we have aliasing distortion. “Typical” bandlimited signal 11/10/2004 ECE-2025 Fall-03 jMc 22 Reconstruction of x(t) xs (t) = x(nTs )δ (t − nTs ) n=−∞ ∞ ∑ Xs ( jω) = 1 Ts X( j(ω k=−∞ ∞ ∑ − kωs )) Xr ( jω) = Hr ( jω)Xs ( jω ) 11/10/2004 ECE-2025 Fall-03 jMc 23 Reconstruction: Frequency-Domain )()()( so overlap,not do )( of copies the,2 If ωωω ω ωω jXjHjX jX srr bs = > Hr ( jω ) 11/10/2004 ECE-2025 Fall-03 jMc 24 Ideal Reconstruction Filter hr (t) = sin πTs t π Ts t Hr ( jω) = Ts ω < π Ts 0 ω > π Ts      hr (0) = 1 hr (nTs ) = 0, n = ±1,±2,… 11/10/2004 ECE-2025 Fall-03 jMc 25 Signal Reconstruction xr (t) = hr (t) ∗ xs (t) = hr (t)∗ x(nTs )δ (t − nTs ) n=−∞ ∞ ∑ xr (t) = x(nTs ) sin πTs (t − nTs ) π Ts (t − nTs )n=−∞ ∞ ∑ Ideal bandlimited interpolation formula xr (t) = x(nTs )hr (t − nTs ) n=−∞ ∞ ∑ 11/10/2004 ECE-2025 Fall-03 jMc 26 Shannon Sampling Theorem “SINC” Interpolation is the ideal PERFECT RECONSTRUCTION of BANDLIMITED SIGNALS 11/10/2004 ECE-2025 Fall-03 jMc 27 Reconstruction in Time-Domain 11/10/2004 ECE-2025 Fall-03 jMc 28 Ideal C-to-D and D-to-C x[n] = x(nTs ) xr (t) = x[n] sin πTs (t − nTs ) π Ts (t − nTs )n=−∞ ∞ ∑ Ideal Sampler Ideal bandlimited interpolator Xr ( jω) = Hr ( jω)Xs ( jω )Xs ( jω) = 1 Ts X( j(ω k=−∞ ∞ ∑ − kωs ))