Digital signal processing

[...]... Sampling a signal in the time domain corresponds in the frequency domain to convolving its spectrum with a Dirac comb The resulting copies of the original signal spectrum in the spectrum of the sampled signal are called “images” 48 Nyquist limit and anti-aliasing filters If the (double-sided) bandwidth of a signal to be sampled is larger than the sampling frequency fs , the images of the signal that... fs of the initial sampled signal by a factor N before passing it to the digital- to-analog converter While this requires more CPU operations and a faster D/A converter, the requirements on the subsequently applied analog reconstruction filter are much less stringent Explain why, and draw schematic representations of the signal spectrum before and after all the relevant signal- processing steps Exercise... 31–44 Hz, which the MATLAB Signal Processing Toolbox function cheby2(3, 30, [31 44]/1500) will design for you • Then sample the resulting signal at 30 Hz by setting all but every 100-th sample value to zero • Generate with cheby2(3, 20, [30 45]/1500) another band-pass filter for the interval 30–45 Hz and apply it to the above 30-Hzsampled signal, to reconstruct the original signal (You’ll have to multiply... of a sampled signal A signal x(t) that is sampled with frequency fs has a spectrum that is periodic with a period of fs x(t) ˆ s(t) x(t) · 0 = t t −1/fs 0 1/fs −1/fs 0 1/fs ∗ 0 f t ˆ X(f ) S(f ) X(f ) = −fs fs f −fs 0 fs f 60 Continuous vs discrete Fourier transform • Sampling a continuous signal makes its spectrum periodic • A periodic signal has a sampled spectrum We sample a signal x(t)... their signal attenuation gradually increases The sampling frequency is therefore usually chosen somewhat higher than twice the highest frequency of interest in the continuous signal (e.g., 4×) On the other hand, the higher the sampling frequency, the higher are CPU, power and memory requirements Therefore, the choice of sampling frequency is a tradeoff between signal quality, analog filter cost and digital. .. reconstruction of the original continuous signal by removing the aliasing frequencies with a reconstruction filter Therefore, it is advisable to limit the bandwidth of the input signal to the sampling frequency fs before sampling, using an anti-aliasing filter In the common case of a real-valued base-band signal (with frequency content down to 0 Hz), all frequencies f that occur in the signal with non-zero power should... Modulation with a carrier frequency fc shifts the spectrum of a signal x(t) into the desired band Amplitude modulation (AM): y(t) = A · cos(2πtfc ) · x(t) X(f ) Y (f ) ∗ −fl 0 fl f = −fc fc f −fc 0 fc f The spectrum of the baseband signal in the interval −fl < f < fl is shifted by the modulation to the intervals ±fc − fl < f < ±fc + fl How can such a signal be demodulated? 44 Sampling using a Dirac comb The... during sampling.) • Plot all the produced sequences and compare the original band-pass signal and that reconstructed after being sampled at 30 Hz Why does the reconstructed waveform differ much more from the original if you reduce the cut-off frequencies of both band-pass filters by 5 Hz? 58 Spectrum of a periodic signal A signal x(t) that is periodic with frequency fp can be factored into a single period... a sampled baseband signal: • Generate a one second long Gaussian noise sequence {rn } (using MATLAB function randn) with a sampling rate of 300 Hz • Use the fir1(50, 45/150) function to design a finite impulse response low-pass filter with a cut-off frequency of 45 Hz Use the filtfilt function in order to apply that filter to the generated noise signal, resulting in the filtered noise signal {xn } • Then... all the relevant signal- processing steps Exercise 8 Similarly, explain how oversampling can be applied to lessen the requirements on the design of an analog anti-aliasing filter 55 Band-pass signal sampling Sampled signals can also be reconstructed if their spectral components remain entirely within the interval n · fs /2 < |f | < (n + 1) · fs /2 for some n ∈ N (The baseband case discussed so far is just . (£40) → K. Steiglitz: A digital signal processing primer – with appli- cations to digital audio and computer music. Addison-Wesley, 1996. (£40) → Sanjit K. Mitra: Digital signal processing – a computer-based approach ed., Prentice-Hall, 1999. (£47) → J. Stein: Digital signal processing – a computer science per- spective. Wiley, 2000. (£74) → S.W. Smith: Digital signal processing – a practical guide for engineers. standards for audio- visual signals 9 Textbooks → R.G. Lyons: Understanding digital signal processing. Prentice- Hall, 2004. (£45) → A.V. Oppenheim, R.W. Schafer: Discrete-time signal process- ing.