Given a bandpass signal with a spectrum shown in Fig. 11.46,...

Given a bandpass signal with a spectrum shown in Fig. 11.46, and assuming $f_s=10 \mathrm{kHz}$, select the sampling rate and sketch the sampled spectrum ranging from $0 \mathrm{~Hz}$ to the carrier frequency for each of the following carrier frequency $f_c$ (a) $f_c=15 \mathrm{kHz}$ (b) $f_c=20 \mathrm{kHz}$

Given a bandpass signal with a spectrum shown in Fig. 11.46, and assuming $f_s=10 \mathrm{kHz}$, select the sampling rate and sketch the sampled spectrum ranging from $0 \mathrm{~Hz}$ to the carrier frequency for each of the following carrier frequency $f_c$
(a) $f_c=15 \mathrm{kHz}$
(b) $f_c=20 \mathrm{kHz}$

Key Concepts

Carrier Frequency Considerations

The carrier frequency sets the central position of the bandpass signal's spectrum. Different carrier frequencies shift the spectrum along the frequency axis, which influences the positioning of spectral replicas after sampling. Understanding how the carrier frequency affects the placement of these replicas is key to selecting a proper sampling rate and correctly interpreting the resulting sampled spectrum.

Aliasing and Spectral Replication

When a signal is sampled, its spectrum is replicated at intervals equal to the sampling rate. For bandpass signals, these replicated spectra must not overlap in the frequency domain; otherwise, aliasing occurs. Proper analysis of these spectral replicas allows one to determine the appropriate sampling rate to ensure that the desired band remains isolated for accurate signal reconstruction.

Sampling Rate and the Nyquist Criterion

The sampling rate is the frequency at which a continuous signal is measured, and according to the Nyquist criterion, it must be at least twice the highest frequency present in the signal to avoid aliasing. In the context of bandpass signals, the effective bandwidth rather than the absolute highest frequency is used to guide the choice of a sampling rate, which can lead to practical reductions in the required sampling rate if the band is narrow.

Bandpass Sampling

Bandpass sampling is a technique used to sample signals whose energy is concentrated in a band away from zero frequency. By properly selecting the sampling rate based on the band’s width and location, one can avoid aliasing while potentially using a sampling rate lower than the Nyquist rate associated with the highest frequency component.

Transcript

00:01 Design a series rlc type band pass filter with the cutoff frequencies 10 khz and 11 kilohertz, assuming that the capacitance is 80 picofarad find rl and q.

00:25 The frequencies omega 1 less than omega -0, that is resonant frequencies is less than omega -2.

00:34 Here, omega 1 is the lower cutoff frequency, omega 2 is the upper cutoff frequency, and omega not is the center frequency.

00:44 For the series rlc circuit, the transfer function h -oega equals the output voltage by input can be written as r by r plus j -omega -l plus 1 by j -o -mega -c.

00:59 Taking out common r -by -l, it will be j -o -mega -1 -l -c.

01:07 Minus omega square plus j into r by l omega.

01:14 Omega.

01:15 Calculating the bandwidth of the filter, which is the difference of lower and upper cutoff frequencies, that is 2 pi, f2 minus 2 pi, f1, which is equals to 2 pi into 11 into 10 raised to the power 3 minus 10 into 10 raised to the power 3.

01:35 2 pi into 10 raise to the power 3 is the bandwidth.

01:39 The relation between center frequency and the side band frequency omega 0 is omega 1 plus omega 2 by 2.

01:49 Rewriting the above equation, f0 equals f1 plus f2 by 2.

01:55 Substitute the value and calculate the center frequency, omega 0 equals 2 pi f0.

02:06 The f0 value here is 10 plus 11 10 plus 11 into 10 raised to the power 3 divided by 2.

02:17 Which is equal to 20, the center frequency is 10 .5 kilohertz.

02:30 Substitute here the f0 value...