SP_VI_P01_4p - Introduction to Signal Processing 1...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Signal Processing 1 Copyright © 2005-2009 – Hayder Radha VIII. Sampling: The Bridge From Continuous To Discrete A. The Sampling Theorem A signal with bandwidth Hz B ( () 0 X ω = for 2 B π > ) can be reconstructed exactly (without error) from its samples taken uniformly at rate 2 s f B > samples per seconds. Introduction to Signal Processing 2 Copyright © 2005-2009 – Hayder Radha Consider a signal () x t with bandwidth : Hz f B = or 2 rad/ sec B = . Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha -1 0 1 2 3 4 5 6 -0.5 0 0.5 1 A B 0 2 π B -2 π B ω X( ω ) f (Hz) x t t Introduction to Signal Processing 4 Copyright © 2005-2009 – Hayder Radha x t is sampled by multiplying it with an impulse train function T t δ consisting of impulses at time periods T . 0 1 2 3 4 5 6 0 1 1.5 2 2.5 3 Sampler Ideal lowpass filter, cutoff B Hz t T t T t xt x t T
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Introduction to Signal Processing 5 Copyright © 2005-2009 – Hayder Radha The frequency of the sampling impulse train function is 1 s f T = . The schematic shows the entire sampler system. The sampled signal () x t consists of a series of impulses. The n -th impulse at time nT has magnitude x nT . () () ( ) ( ) T n x tx t t x n Tt n T δδ == Introduction to Signal Processing 6 Copyright © 2005-2009 – Hayder Radha T t δ is a periodic signal and has trigonometric series, [] 1 1 2cos 2cos2 T sss tt t t T ωωω =+ + + + " where 2 2 ss f T π ω . Introduction to Signal Processing 7 Copyright © 2005-2009 – Hayder Radha Therefore, T x t t = . 2()c o s 1 2 ()cos2 o s3 s s s xt t x T x + ⎡⎤ ⎢⎥ + + ⎣⎦ " Introduction to Signal Processing 8 Copyright © 2005-2009 – Hayder Radha -1 0 1 2 3 4 5 6 -0.5 0 0.5 1 t x t
Background image of page 2
Introduction to Signal Processing 9 Copyright © 2005-2009 – Hayder Radha To find () X ω we find the Fourier transform of x t term- by-term, [ ] () 2 ()cos 1 2()c o s2 o s3 s ss xt t t t T ωω = + ⎡⎤ = ⎢⎥ ++ + ⎣⎦ " F 1 st term: [ ] ( ) X = 2 nd term: [ ] o s ( ) ( ) s t X X =− 3 rd term: [ ] o ( 2 ) ( s t X X =−++ Introduction to Signal Processing 1 0 Copyright © 2005-2009 – Hayder Radha Etc. 1 ( ) s n X Xn T =−∞ Recovery of the original signal x t (or X ) from its sampled version x t (or X ) is only possible if there is no overlap between successive cycles in X . From the figure we see this will hold true only when, 2 s f B > or 1 2 T B < . Introduction to Signal Processing 1 1 Copyright © 2005-2009 – Hayder Radha The minimum sampling rate 2 s f B = required for signal reconstruction is also called the Nyquist rate . Introduction to Signal Processing 1 2 Copyright © 2005-2009 – Hayder Radha Example 8.1: This example illustrates the effect of sampling a signal at a frequency: below the Nyquist rate ( undersampling ) and above the Nyquist rate ( oversampling ). Consider a signal 2 () s inc(5 ) x tt π = with spectrum () 0 . 2( /2 0) X ωπ .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Introduction to Signal Processing 1 3 Copyright © 2005-2009 – Hayder Radha -0.5 0 0.5 0.2 0.4 0.6 0.8 1 5Hz(10 rad/ sec) B π = The Nyquist rate is 10Hz 1/10 0.1sec s f T =⇒ = = .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 16

SP_VI_P01_4p - Introduction to Signal Processing 1...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online