Lecture 1. 27 Signals with limited frequency range Lecture 1.27 Signals with limited frequency range. Recovery signals discrete samples.

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

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

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, …

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

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

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

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

Sampling at the Nyquist rate x * = = f(t) S(ω)

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

Sampling below the Nyquist rate x * = = f(t) S(ω)

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

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

Undersampled Reconstruction Reconstruction error Original Signal Undersampled Reconstruction

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

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

Rectangle Reconstruction Reconstruction error Original Signal Rectangle Reconstruction

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

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

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

Nyquist sampling rate for low-pass and bandpass signals

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.

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.