Chapter 3 Sampling

Introduction Digital signal processing system Most signals in nature are in analog form Needs an analog-to-digital conversion process Sampling Acquisition of continuous signal at discrete-time intervals Conversion of continuous signal to discrete-time signal A-D convertor Digital system Fig. 3-1.

Sampling theory Sampling with interval T Sampling with a periodic impulse train converts to a discrete-time sequence Discrete signal Analog signal Sampler Fig. 3-2.

Same sampled valued at each sampling point Fig. 3-3.

Mathematical representation of sampling theorem Multiplying continuous signal with impulse train Fig. 3-4.

Sampled signal in time domain Impulse signal property gives: Applying it to Eq.(3-1) (3-1) (3-2) (3-3)

Using convolution properties of Fourier transform By example 2-8 Convolution in frequency domain Shifting the signal to each position of impulses (3-4) (3-5) (3-6)

Sampling in frequency domain Repeats the spectrum of sampled signal at period of Scaled by If , each repeated signal is preserved If , each repeated signal is overlapped Fig. 3-5.

Sampling sinusoidal signal of frequency, with three sampling intervals Fig. 3-6.

Sampling frequency required to reconstruct original signal Nyquist frequency (3-7) (3-8)

Reconstruction of continuous signal from samples Reconstruction of continuous signal from sampled signal Using low-pass filter in time domain By Eq.(3-3) Interpolation with ideal low-pass filter Impulse response (3-9) (3-10)

Reconstructed continuous signal (3-11) Fig. 3-7.

Reconstructed signal with Using convolution of sinc function (3-12) (3-13) Fig. 3-8.