lecture 3+4.pdf
- 1. Aliasing 24 How many hertz can the human eye see? • Most don't notice unless it is under 50 or 60 Hz. • Generally, people notice when the frame-rate is less than the refresh rate of the display. • Depending on the type of CRT, you couldn't see flicker at 30 Hz or you could still see it at 120 Hz.
- 2. Aliasing • When the minimum sampling rate is not respected, distortion called aliasing occurs. • Aliasing causes high frequency signals to appear as lower frequency signals. • To be sure aliasing will not occur, sampling is always preceded by low pass filtering. • The low pass filter, called the anti-aliasing filter, removes all frequencies above half the selected sampling rate. 25
- 3. Aliasing • Figure illustrates sampling a 40 Hz sinusoid • The sampling interval between sample points is T = 0.01 second, and the sampling rate is thus fs = 100 Hz. • The sampling theorem condition is satisfied 26
- 4. Aliasing • Figure illustrates sampling a 90 Hz sinusoid • The sampling interval between sample points is T = 0.01 second, and the sampling rate is thus fs = 100 Hz. • The sampling theorem condition is not satisfied 27
- 5. Aliasing 28
- 6. Sampling Effect in Time Domain 29 Example of Aliasing in the time domain of various sinusoidal signals ranging from 10 kHz to 80 kHz with a sampling frequency Fs = 40 kHz.
- 7. Time & Frequency Domains 30 • There are two complementary signal descriptions. • Signals seen as projected onto time or frequency domains.
- 8. Time & Frequency Domains 31
- 10. Frequency Range of Analog & Digital Signals • For analog signals, the frequency range is from -∞ Hz to ∞ Hz • For digital signals, the frequency range is from 0 Hz to Fs/2 Hz 33
- 11. Sampling Effect in Frequency Domain • Sampling causes images of a signal’s spectrum to appear at every multiple of the sampling frequency fs. • For a signal with frequency f, the sampled spectrum has frequency components at kfs ± f 34
- 12. 35 Sampling Effect in Frequency Domain
- 13. 36 Sampling Effect in Frequency Domain
- 14. Anti Aliasing Filter • A signal with no frequency component above a certain maximum frequency is known as a band-limited signal. • In our case we want to have a signal band-limited to ½ Fs. • Some times higher frequency components (both harmonics and noise) are added to the analog signal (practical signals are not band-limited). • In order to keep analog signal band-limited, we need a filter, usually a low pass that stops all frequencies above ½ Fs. • This is called an “Anti-Aliasing” filter. 37
- 15. Anti Aliasing Filter • Anti-aliasing filters are analog filters. • They process the signal before it is sampled. • In most cases, they are also low-pass filters unless band-pass sampling techniques are used. 38
- 16. Under Sampling • If the sampling rate is lower than the required Nyquist rate, that is fS < 2W, it is called under sampling. • In under sampling images of high frequency signals erroneously appear in the baseband (or Nyquist range) due to aliasing. 39
- 17. Sampling of Band Limited Signals Signals whose frequencies are restricted to a narrow band of high frequencies can be sampled at a rate similar to twice the Bandwidth (BW) instead of twice the maximum frequency. Fs ≥ BW 40
- 18. Sampling of Band Limited Signals • While this under-sampling is normally avoided, it can be exploited. • For example, in the case of band limited signals all of the important signal characteristics can be deduced from the copy of the spectrum that appears in the baseband through sampling. • Depending on the relationship between the signal frequencies and the sampling rate, spectral inversion may cause the shape of the spectrum in the baseband to be inverted from the true spectrum of the signal. 41
- 19. Sampling of Band Limited Signals 42 Figure: Signal recovered From Nyquist range are Base band versions of the Original signal. Sampling rate is Important to make sure no aliasing and spectral inversion occurs. (a) Fs = 80 kHz, signal spectrum is Inverted in the baseband. (b) Fs = 100 kHz, the lowest Frequencies In the signal alias to the highest frequencies. (c) Fs = 120 kHz, No spectral Inversion occurs.
- 20. Over Sampling • Oversampling is defined as sampling above the minimum Nyquist rate, that is, fS > 2fmax. • Oversampling is useful because it creates space in the spectrum that can reduce the demands on the analog anti- aliasing filter. 43
- 21. Over Sampling • In the example below, 2x oversampling means that a low order analog filter is adequate to keep important signal information intact after sampling. • After sampling, higher order digital filter can be used to extract the information. 44
- 22. Over Sampling • The ideal filter has a flat pass-band and the cut-off is very sharp, since the cut-off frequency of this filter is half of that of the sampling frequency, the resulting replicated spectrum of the sampled signal do not overlap each other. Thus no aliasing occurs. • Practical low-pass filters cannot achieve the ideal characteristics. • Firstly, this would mean that we have to sample the filtered signals at a rate that is higher than the Nyquist rate to compensate for the transition band of the filter 45
- 23. Spectra of Sampled signals 46 Figure: Signal ‘s Spectra (i) Over sampled (ii) Nyquest Rate (iii) Under Sampled
- 24. Sampling Low Pass Signals 47
- 25. Exercise 48 Exercise-1: If the 20 kHz signal is under-sampled at 30 kHz, find the aliased frequency of the signal. Exercise-2: A voice signal is sampled at 8000 samples per second. i. What is the time between samples? ii. What is the maximum frequency that will be recovered from the signal? Exercise-3: An analog Electromyogram (EMG) signal contains useful frequencies up to 3000 Hz. i. Determine the minimum required sampling rate to avoid aliasing. ii. Suppose that we sample this signal at a rate of 6500 samples/s. what is the highest frequency that can be represented uniquely at this sampling rate?
- 26. Exercise 49 Exercise-4: Humans can hear sounds at frequencies between 0 and 20 kHz. What minimum sampling rate should be chosen to permit perfect recovery from samples? Exercise-5: An ECG signal is sampled at 250 samples per second. i. What is the time between samples? ii. What is the maximum frequency that will be recovered from the signal? Exercise-6: An ultrasound signal ranging in frequency from 900 kHz to 900.5 kHz is under-sampled at 200 kHz. If a 200 Hz target appears in the baseband, what is the actual frequency of the target?
- 27. Exercise-1: If the 20 kHz signal is under-sampled at 30 kHz, find the aliased frequency of the signal. Solution: The Aliased frequency = fs - fmax = 30 kHz – 20 kHz = 10 kHz Exercise-2: A voice signal is sampled at 8000 samples per second. i. What is the time between samples? ii. What is the maximum frequency that will be recovered from the signal? Solution: i. Time between samples = 1/fs = 1/8000 = 0.125 msec
- 28. ii. The maximum frequency that will be recovered from the signal = fs/2 = 4000 Hz. Exercise-3: An analog Electromyogram (EMG) signal contains useful frequencies up to 3000 Hz. i. Determine the minimum required sampling rate to avoid aliasing. ii. Suppose that we sample this signal at a rate of 6500 samples/s. what is the highest frequency that can be represented uniquely at this sampling rate? Solution: i. The minimum required sampling rate to avoid aliasing = 2 x fmax = 2 x 3000 = 6000 Hz ii. The highest frequency that can be represented uniquely at 6500 samples/s = fs/2 = 6500/2 = 3250
- 29. Exercise-4: Humans can hear sounds at frequencies between 0 and 20 kHz. What minimum sampling rate should be chosen to permit perfect recovery from samples? Solution: Minimum sampling rate = 2 x 20 kHz = 40 kHz Exercise-5: An ECG signal is sampled at 250 samples per second. i. What is the time between samples? ii. What is the maximum frequency that will be recovered from the signal? Solution: i. Time between samples = 1/250 = 4 msec ii. The maximum frequency that will be recovered from the signal = fs/2 = 125 Hz