DSP Unit 3 PDF

**UNIT-3** **Bandpass Signals and Sampling:** Bandpass signals are signals whose frequencies are centered around a specific band of frequencies within the spectrum. They are characterized by having a lower cutoff frequency fLf\_LfL​ and an upper cutoff frequency fHf\_HfH​, defining the bandwidth B...

**UNIT-3** **Bandpass Signals and Sampling:** Bandpass signals are signals whose frequencies are centered around a specific band of frequencies within the spectrum. They are characterized by having a lower cutoff frequency fLf\_LfL​ and an upper cutoff frequency fHf\_HfH​, defining the bandwidth B=fH−fLB = f\_H - f\_LB=fH​−fL​. When dealing with bandpass signals in the context of sampling, several considerations come into play: 1. **Sampling Theorem (Nyquist Criterion):** For bandpass signals, the sampling rate fsf\_sfs​ must be at least twice the bandwidth BBB of the signal to avoid aliasing. Mathematically, this is expressed as fs≥2Bf\_s \\geq 2Bfs​≥2B. 2. **Baseband Equivalent Representation:** Bandpass signals can be represented in a baseband equivalent form, which simplifies their analysis and processing. This involves shifting the signal from its original frequency band to a lower frequency range (typically near DC) using techniques such as complex envelope representation or analytic signal representation. 3. **Sampling Bandpass Signals:** To properly sample a bandpass signal without aliasing, one approach is to first down-convert it to a lower frequency (baseband) using mixing techniques with a local oscillator at the carrier frequency fcf\_cfc​. The resulting baseband signal, which now spans from −B/2-B/2−B/2 to B/2B/2B/2, can then be sampled at a rate fsf\_sfs​ satisfying the Nyquist criterion for baseband signals. 4. **Practical Considerations:** In practice, implementing bandpass sampling involves careful design of anti-aliasing filters before sampling and reconstruction filters after sampling. These filters help in ensuring that only the desired band of frequencies is captured and reconstructed accurately without distortion or aliasing. 5. **Applications:** Bandpass signals and their sampling are crucial in various applications such as communication systems (where signals are modulated around a carrier frequency), biomedical signal processing, radar systems, and many more where signals of interest are localized within a specific frequency band. Understanding bandpass signals and their sampling is fundamental for designing efficient and accurate signal processing systems, ensuring that the information carried by these signals is faithfully preserved and utilized in various technological applications. A signal x(t) is a band pass signal and the frequency spectrum of the signal is a shown in figure The signal is having a frequency component in the range of B₁ \< Fe\< B2. The B₂ is the maximum frequency component and according to the Nyquist Theorem the signal should be sample to the twice of maximum frequency component. i.e. 2B to get Alias free signal. Therefore is above signal we need to sample the signal at 2B sample per second. But in the previous discussion we have shown that the band pass signal can be converted into low pass signal by frequency shift Fc could be, \ [\$\$FC = \\ \\frac{B1 + B2}{2}\$\$]{.math.display}\ and sampling we equivalent signal. Such frequency ship can be achieved by multiplying the band pass signal. x(t) = x(t) cos (2[*π*]{.math.inline}fct) - y(t) sin (2[*π*]{.math.inline}fct) by quadrature carriers i.e., cos 2nfet and sin 2xft and after low pass filtering the product to eliminate the signal component at 2 Fc. analog domain and then it passed the filter and being sampled. after sampled has a bandwidth being 2, where B = B\_{2} - B\_{1}. As shown in figure 3.6 the output of filters are sampled at a rate of B sample per second each and the resulting rate is 2B sample per As shown in figure 3.6 the band pass signal is being multiplied by cos 2nFct and sin 2xfct which are quadrature components and being filter with the help of low pass filter to shift the frequency spectrum and A/D converter samples the output of low pass filter at rate of B samples per second.  Analog domain and then it passed the filter and being sampled. After sampled has a band width being 2, where B = B\_{2} - B\_{1}. As shown in figure the output of filters are sampled at a rate of B sample per second each and the resulting rate is 2B sample per As shown in figure the band pass signal is being multiplied by cos 2nFct and sin 2xfct which are quadrature components and being filter with the help of low pass filter to shift the frequency spectrum and A/D converter samples the output of low pass filter at rate of B samples per second. Suppose the upper frequency Fc + B / 2 is multiple of bandwidth B i.e. Fc + [\$\\frac{B}{2}\$]{.math.inline}= KB. for the samples to be odd put n = 2m - 1 in equation (1),we can reduces to  Therefore, we can say that the even numbered sample of x(t) which occur at B sample per second, produces samples by lowpass signal component x(t) and the odd number samples of low pass component y(t). The signal component x(t) and y(t) can be used to reconstruct the equivalent lowpass signal. Thus according to sampling theorem for lowpass signal with T1 = 1 / B i.e 2T = 1 / B  Top of Form Bottom of Form **\ Representation of Bandpass Signals:**