Signals and systems Fall 2003 Lecture #13 21 October 2003 The Concept and representation of periodic Sampling of a ct signal Analysis of Sampling in the Frequency Domain 3. The Sampling Theorem -the Nyquist Rate 4. In the Time Domain Interpolation Undersampling and Aliasing
1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling in the Frequency Domain 3. The Sampling Theorem — the Nyquist Rate 4. In the Time Domain: Interpolation 5. Undersampling and Aliasing Signals and Systems Fall 2003 Lecture #13 21 October 2003
SAMPLING We live in a continuous-time world most of the signals we encounter are CT signals, e.g. x(t). How do we convert them into DT signals x[n? Sampling, taking snap shots of x(t)every Second T-sampling period xn=x(nn,n=.-1,0, 1, 2,.-regularly spaced samples Applications and examples Digital Processing of signals Strobe mages in Newspapers Sampling oscilloscope How do we perform sampling?
We live in a continuous-time world: most of the signals we encounter are CT signals, e.g. x ( t). How do we convert them into DT signals x[n]? SAMPLING How do we perform samplin g ? — Sampling, taking snap shots of x ( t) every T seconds. T – sampling period x[n] ≡ x (nT), n = ..., -1, 0, 1, 2, ... — regularly spaced samples Applications and Examples — Digital Processing of Signals — Strobe — Images in Newspapers — Sampling Oscilloscope — …
Why/when Would a Set of samples Be adequate? Observation: Lots of signals have the same samples X3()x1()x2( -3T -2T by sampling we throw out lots of information all values of x(t) between sampling points are lost Key Question for Sampling Under what conditions can we reconstruct the original CT signal x(t) from its samples
Why/When Would a Set of Samples Be Adequate? • Observation: Lots of signals have the same samples • By sampling we throw out lots of information – all values of x ( t) between sampling points are lost. • Key Question for Sampling: Under what conditions can we reconstruct the original CT signal x ( t) from its samples?
Impulse Sampling- Multiplying x(t) by the sampling function p(t)=∑6(t-n) xp()=(1)=∑(106t-nn)=∑a(nm)6(t-n) n=-∞ n=-0o p() X(tH Xp(t) X() p() T X(0) Xp(t)
Impulse Sampling — Multiplying x(t) by the sampling function
Analysis of sampling in the Frequency domain Multiplication Property Xp(jw)=oX(w)* plow) 2丌 P)=r∑6a-ka) k 2丌 T= Sampling Frequency Important to note:ocl/T )=7∑x)*6(-ka) k=-∞ ∑X(a-k)
Analysis of Sampling in the Frequency Domain Import ant to note: ωs∝1/ T