Discrete-time Signals & Systems

Transcription

1 Discrete-time Signals & Systems S Wongsa Dept. of Control Systems and Instrumentation Engineering, KMU JAN, Overview Signals & Systems Continuous & Discrete ime Sampling Sampling in Frequency Domain Sampling heorem Aliasing & Anti-Aliasing Filter 2

2 Lecture plan Lecture Date opic 1 12 & 13 Jan 09 Discrete-time signals and systems; Sampling of continuous-time signals 2 19 & 20 Jan 09 Discrete-ime Fourier ransform (DF)& Discrete-Fourier ransform (DF) 3 26 & 27 Jan 09 Fast Fourier ransform (FF)& Applications (Lab I) 4 2 & 3 Feb 09 z-ransform 5 9 & 10 Feb 09 ransform-domain analysis and LI systems 6 16 & 17 Feb 09 Discrete-time system analysis Lab (II) 7 23 & 24 Feb 09 utorial Course website: 3 Recommended extbooks 1. Fundamentals of Signals and Systems Using MALAB, Edward W. Kamen and Bonnie S. Heck, Prentice Hall International Inc. 2. Discrete-time signal processing, A.V. Oppenheim, R.W. Schafer, and J. R. Buck, 2nd edition, Prentice Hall, Signals and Systems, Alan v. Oppenheim et al., 2nd Edition, Prentice Hall. 4. Signals and Systems, Simon Haykin & Barry Van Veen, 2nd edition, Wiley,

3 Signals & Systems A signal is a varying phenomenon that can be measured. A system responses to particular signals by producing other signals. Source: Signals & Systems, MI, Fall Signals & Systems An image is also a signal! Source: Yao Wang, Introduction, Review of Signals & Systems, Image Quality Metrics, Polytechnic University, Brooklyn, NY 6

4 Continuous & Discrete ime Most of the signals in the physical world are C signals, e.g. voltage & current, pressure, temperature, velocity, etc. But digital computations are done in discrete time. Source: Signals & Systems, MI, Fall Discrete-time processing of continuous-time signals Sampling Reconstruction C/D = continuous-to-discrete D/C = discrete-to-continuous Zero-order-hold (ZOH) Source: Signals & Systems, MI, Fall

5 Sampling Sampling is the process of getting a discrete signal from a continuous one. It enables the processing of signal by digital computer. x (t) (t) x s Discrete-time signal x s ( t) = x( n ) = x[ n], n= 0, ± 1, ± 2,K where is a sampling time. 9 Sampling We would like to sample in a way that preserves information, which may not seem possible because information between samples is lost. How can we minimise the distortion of reconstructed signal? Source: Signals & Systems, MI, Fall

6 Sampling x(t) x s (t) δ (t) X where x s ( t) = x( t) δ ( t) n= δ ( t ) = δ ( t n ) 11 Sampling in frequency domain he Fourier transform of x s (t) : where 1 X s ( ω) = X ( ω kωs ) k= 2π ωs = is the sampling frequency in rad/sec. If x(t) has bandwidth B and if ωs > 2B 12

7 Bandlimited reconstruction he high frequency copies can be removed with a low-pass and then multiplying by to undo the amplitude scaling. 13 Sampling theorem A signal with bandwidth B can be reconstructed completely and exactly from sampled signal by lowpass filtering with cutoff frequency B if s > 2B. ω ω s = 2B is called the Nyquist sampling frequency / Nyquist rate. 14

8 Aliasing Q : What if the signal is not bandlimited nor is the sampling frequency greater than the Nyquist sampling frequency? A : he high frequency components of x(t) will be transposed to low-frequency components, leading to a phenomenon called aliasing. (ω) X s Aliasing Aliasing -B B 15 Aliasing What are the consequences of aliasing? - a distorted version of the original signal x(t). - it makes two continuous sinusoids of different frequencies indistinguishable when sampled. 3 2 Amplitude ime (sec) Aliasing: a 52 Hz sinusoid sampled at 50 Hz. 16

9 Aliasing Aliasing: a 52 Hz sinusoid sampled at 50 Hz. 3 2 Amplitude ime (sec) Magnitude Frequency (Hz) 17 Anti-Aliasing Filter o avoid aliasing, restrict the bandwidth of a signal to approximately satisfy the sampling theorem by removing frequency components that alias before sampling. 18

10 Review Questions 1. If we used x(t) below and sampled it at 20 khz, how many samples would we have after 60 ms? x( t) = 3cos(2π 404t+ π / 4) + 2cos(2π 6510) + cos(2π 660t π / 5) 2. x(t) = 2 cos(2π700t 5π/2) + 3 cos(2π450t) + cos(2π630t + 2π/5) What is the minimum sampling rate for this signal? 3. A periodic signal with a period of 0.1 ms is sampled at 44 khz. o what frequency does the eighth harmonic alias? 19