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 4 & 5 Jan 11 Discrete-time signals and systems; Sampling of continuous-time signals 2 11 & 12 Jan 11 Discrete-ime Fourier ransform (DF) & Discrete-Fourier ransform (DF) 3 18 & 19 Jan 11 Fast Fourier ransform (FF) & Applications (Lab I) 4 25 & 26 Jan 11 z-ransform 5 1 & 2 Feb 11 ransform-domain analysis and LI systems 6 8 & 9 Feb 11 Discrete-time system analysis Lab (II) 7 15 & 16 Feb 11 utorial Course website: 3 Grading 1.) Graded homework worth 10% 2.) Laboratory assignments worth 10% 3.) Final MALAB exam worth 5% 4.) Final exam worth 25% 4

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, 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

4 Signals & Systems An image is also a signal! Source: Yao Wang, Introduction, Review of Signals & Systems, Image Quality Metrics, Polytechnic University, Brooklyn, NY 7 Discrete-time processing of continuous-time signals Sampling Reconstruction e.g. DSP, Controller etc. 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

5 Discrete-time processing of continuous-time signals Source: Prof. Mark Fowler, EECE 301 Signals & Systems, Binghamton University. 9 Discrete-time processing of continuous-time signals Sampling Reconstruction e.g. DSP, Controller etc. 10

6 DSP for Detection of Weld Defects Defects? Original image After DSP Porosity Incomplete penetration Source: W.Yuttiwat et al., Visual Inspection of Weld Defects by Radiography Image Processing, IE Network DSP: Biomedical Imaging X-Ray C MRI Magnetic Resonance Imaging make use of radiation to get an internal view of the body. be blocked by some form of dense tissue, therefore the image quality when looking at soft tissues will be poor. can pose the risk of irradiation. use a series of X-ray beams to create cross-sectional images. DSP is used to generate a 3D image of the internals of an object from a large series of 2D X-ray images taken around a single axis of rotation uses magnetic fields in conjunction with radio waves to give high detail in the soft tissues. No biological hazards have been reported with the use of the MRI. Source:

7 Audio Signal Processing Music Speech Generation e.g. ext-to-speech Synthesis, Voice conversion Speech Recognition Source: Discrete-time processing of continuous-time signals Sampling Reconstruction Source: Signals & Systems, MI, Fall

8 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. 15 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

9 Sampling x(t) (t) x s (t) δ X x ( t) = x( t) δ ( t) s where n= δ ( t ) = δ ( t n ) 17 Sampling in frequency domain he Fourier transform of x s (t) : 1 X s ( ω) = X ( ω kωs ) k= where 2π ωs = is the sampling frequency in rad/sec. Goal: Determine under what conditions we get: Reconstructed C signal = Original C signal 18

10 Sampling in frequency domain If x(t) has bandwidth B and if ωs > 2B x(t) is a bandlimited signal. 1 X s ( ω) = X ( ω kωs ) k= he high frequency copies can be removed with a low-pass and then multiplying by to undo the amplitude scaling. 19 Sampling theorem A bandlimited signal with bandwidth B can be reconstructed completely and exactly from its samples as long as they are taken at rateωs > 2B ωs = 2B is called the Nyquist sampling frequency / Nyquist rate. NB: Sampling at Nyquist rate is only possible if an IDEAL lowpass filter is used. In practice we generally need to choose a sampling rate above the Nyquist rate. 20

11 What if the samples are not taken fast enough? (ω) X s Aliasing Aliasing -B B he high frequency components of x(t) will be transposed to low-frequency components, leading to a phenomenon called aliasing. 21 What if the signal is not bandlimited? For non-bandlimited signal aliasing always happens regardless of ω s value. 22

12 Aliasing What are the consequences of aliasing? - 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. 23 Aliasing What are the consequences of aliasing? - a distorted version of the original signal x(t). Example: original music sampled at 44.1kHz (CD-quality) he at 4kHz downsampled version. 24

13 Anti-Aliasing Filter o avoid aliasing, in practice we use a C lowpass filter before the ADC to restrict the bandwidth of a signal to approximately satisfy the sampling theorem. Fs = 44.1 khz Source: Prof. Mark Fowler, EECE 301 Signals & Systems, Binghamton University. 25 Suggested Readings Steven W. Smith, Chapter 3: ADC and DAC, he Scientist and Engineer's Guide to Digital Signal Processing 26

14 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? 27 Summary 28