By Ken Steiglitz
This article is directed on the marketplace of DSP clients caused via the advance of robust and cheap software program instruments to research indications. those instruments enable refined manipulation of signs yet don't offer an figuring out of the idea or the basis for the ideas. This paintings develops an method of the advance of the maths of DSP and makes use of examples from components of the spectrum widespread to newbies, including questions and instructed experiments
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Hence, the analog signal can be perfectly recovered from its sampled version. Let us study the following example. 1 Suppose that an analog signal is given as xðtÞ ¼ 5cosð2p$1; 000tÞ; for t ! 0 and is sampled at the rate 8,000 Hz. a. Sketch the spectrum for the original signal. b. Sketch the spectrum for the sampled signal from 0 to 20 kHz. 2 Signal Reconstruction 21 Solution: a. Since the analog signal is sinusoid with a peak value of 5 and frequency of 1,000 Hz, we can write the sine wave using Euler’s identity: e j2pÂ1;000t þ e Àj2pÂ1;000t Þ ¼ 2:5e j2pÂ1;000t þ 2:5e Àj2pÂ1;000t 5cosð2p Â 1; 000tÞ ¼ 5$ð 2 which is a Fourier series expansion for a continuous periodic signal in terms of the exponential form (see Appendix B).
8. 9. 1 1 1 Xðf þ fs Þ þ Xðf Þ þ Xðf À fs Þ þ / T T T That is, the sampled signal spectrum is a scaled and shifted version of its analog signal spectrum and its replicas centered at the frequencies that are multiples of the sampling rate. The analog anti-aliasing lowpass filter is used before ADC to remove frequency components higher than the folding frequency to avoid aliasing. The reconstruction (analog lowpass) filter is adopted after DAC to remove the spectral images that exist in the sample-and-hold signal and obtain the smoothed analog signal.
29, we see that xmin ¼ 0, xmax ¼ 8D, and m ¼ 3. 22) gives each quantization level as follows: xq ¼ 0 þ iD, i ¼ 0; 1; /; L À 1, where L ¼ 23 ¼ 8 and i is the integer corresponding to the 3-bit binary code. 1 details quantization for each input signal subrange. 30, where we have xmin ¼ À4D, xmax ¼ 4D, and m ¼ 3. 2. 29 Characteristics of the unipolar quantizer. 30 Characteristics for the bipolar quantizer. 9 Assuming that a 3-bit ADC channel accepts analog input ranging from 0 to 5 volts, determine a.
A digital signal processing primer with applications to digital audio and computer music by Ken Steiglitz
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