1. Lecture 1. 27 Signals with limited frequency range
Lecture Signals with limited frequency range. Recovery signals discrete samples.

2. Sampling: Time Domain
Many signals originate as continuous-time signals, e.g. conventional music or voice
By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbers
n  {…, -2, -1, 0, 1, 2,…}
Ts is the sampling period. Ts t Ts
s(t)
Sampled analog waveform
impulse train

3. Sampling: Frequency Domain
Replicates spectrum of continuous-time signal
At offsets that are integer multiples of sampling frequency
Fourier series of impulse train where ws = 2 p fs
Example
Modulation by cos(s t)
Modulation by cos(2 s t) w
F(w)
2pfmax
-2pfmax w
G(w) ws
2ws
-2ws
-ws

4. Shannon Sampling Theorem
A continuous-time signal x(t) with frequencies no higher than fmax can be reconstructed from its samples x[n] = x(n Ts) if the samples are taken at a rate fs which is greater than 2 fmax.
Nyquist rate = 2 fmax
Nyquist frequency = fs/2.
What happens if fs = 2fmax?
Consider a sinusoid sin(2 p fmax t)
Use a sampling period of Ts = 1/fs = 1/2fmax.
Sketch: sinusoid with zeros at t = 0, 1/2fmax, 1/fmax, …

5. Shannon Sampling Theorem
Assumption
Continuous-time signal has no frequency content above fmax
Sampling time is exactly the same between any two samples
Sequence of numbers obtained by sampling is represented in exact precision
Conversion of sequence to continuous time is ideal
In Practice
Continuous-time signal has no frequency content above fmax
Sampling time is exactly the same between any two samples
Quite well executed
Great difficulties in putting into practice

6. Band-limited transform pairs
sinc(x)
sinc2(x)
f(x)
F(s)

7. Spectrum is replicated an infinite number of times
Sampling theory x *
Spectrum is replicated an infinite number of times = =
f(t)
S(ω)

8. Reconstruction theory* x
sinc = =
f(t)
S(ω)

9. Sampling at the Nyquist ratex * = =
f(t)
S(ω)

10. Reconstruction at the Nyquist rate* x = =
f(t)
S(ω)

11. Sampling below the Nyquist ratex * = =
f(t)
S(ω)

12. Reconstruction below the Nyquist rate* x = =
f(t)
S(ω)

13. Effect of Aliasing
Fourier Theorem states that any waveform can be reproduced by sine waves.
Improperly sampled signals will have other sine wave components.

14. Undersampled Reconstruction
Reconstruction error
Original Signal
Undersampled Reconstruction

15. Reconstruction with a triangle function* x = =
f(t)
S(ω)

16. Reconstruction with a rectangle function* x = =
f(t)
S(ω)

17. Rectangle Reconstruction
Reconstruction error
Original Signal
Rectangle Reconstruction

18. Sampling a rectangle x * = =
f(t)
S(ω)

19. Reconstructing a rectangle (jaggies)* x = =
f(t)
S(ω)

20. Summary. Sampling and reconstruction
Aliasing is caused by
Sampling below the Nyquist rate,
Improper reconstruction, or
Both
Fourier theory explains jaggies as aliasing. For correct reconstruction:
Signal must be band-limited
Sampling must be at or above Nyquist rate
Reconstruction must be done with a sinc function

21. Nyquist sampling rate for low-pass and bandpass signals

22. Example 1
A complex low-pass signal has a bandwidth of 200 kHz. What is the minimum sampling rate for this signal?
Solution
The bandwidth of a low-pass signal is between 0 and f, where f is the maximum frequency in the signal. Therefore, we can sample this signal at 2 times the highest frequency (200 kHz). The sampling rate is therefore 400,000 samples per second.

23. Example 2
A complex bandpass signal has a bandwidth of 200 kHz. What is the minimum sampling rate for this signal?
Solution
We cannot find the minimum sampling rate in this case because we do not know where the bandwidth starts or ends. We do not know the maximum frequency in the signal.