0.5 Sampling with automatic gain control (Page 3/19)

Software receiver design Page 3 / 19

The spectrum of a sampled signal is periodic with period equal to f_s. In case (a), the original spectrum W(f) is bandlimited to less than f_s/2 and there is no overlapping of the replicas. When W(f) is not bandlimited to less than f_s/2, as in (b), the overlap of the replicas is called aliasing. — The spectrum of a sampled signal is periodic with period equal to $f_{s}$ . In case (a), the original spectrum $W (f)$ is bandlimited to less than $\frac{f_{s}}{2}$ and there is no overlapping of the replicas. When $W (f)$ is not bandlimited to less than $\frac{f_{s}}{2}$ , as in (b), the overlap of the replicas is called aliasing.

Nyquist Sampling Theorem: If the signal $w (t)$ is bandlimited to $B$ , ( $W (f) = 0$ for all $| f | > B$ ) and if the sampling rate is faster than $f_{s} = 2 B$ , then $w (t)$ can be reconstructed exactly from its samples $w (k T_{s})$ .

On the other hand, in part (b) of [link] , the replicas overlap because the repetitions are narrowerthan the width of the spectrum $W (f)$ . In this case, it is impossible to recover the original spectrum perfectly from thesampled spectrum, and hence it is impossible to exactly recover the original waveform from the sampled version.The overlapping of the replicas and the resulting distortions in the reconstructed waveform are called aliasing .

Bandwidth can also be thought of as limiting the rate at which data can flow over a channel. When a channelis constrained to a bandwidth $2 B$ , then the output of the channel is a signal with bandwidth no greater than $2 B$ . Accordingly, the output can contain no frequencies above $f_{s}$ , and symbols can be transmitted no faster than one every $T_{s}$ seconds, where $\frac{1}{T_{s}} = f_{s}$ .

Human hearing extends up to about 20 KHz. What is the minimum sampling rate needed to fully capture a musicalperformance? Compare this to the CD sampling rate of $44.1$ KHz. Some animal sounds, such as the singing of dolphins andthe chirping of bats, occur at frequencies up to about 50 KHZ. What does this imply about CD recordings of dolphin or bat sounds?

U.S. high-definition (digital) television (HDTV) is transmitted in the same frequency bands as conventional television (for instance,Channel 2 is at 54 MHz), and each channel has a bandwidth of about 6 MHz. What is the minimum sampling rate needed tofully capture the HDTV signal once it has been demodulated to baseband?

An analog signal has nonzero values in its magnitude spectrum at every frequencybetween $- B$ and $B$ . This signal is sampled with period $T$ where $1 / T > B$ . TRUE or FALSE: the discrete-time signal can have components in its spectrumat frequencies between $B$ and $1 / T$ .

The triangularly-shaped magnitude spectrum of a real message signal $w (t)$ is shown in [link] where $B = 0.2$ MHz. The received signal

r (t) = 0.15 w (t) cos (2 π f t)

is modulated by an AM-with-suppressed-carrier with $f = 1.45$ MHz, and attentuated. With $\frac{1}{2 B} < T_{1} < \frac{1}{f}$ , select $T_{1}$ , $T_{2}$ , $T_{3}$ , and $β$ so the magnitude spectrum of $x [k]$ matches the magnitude spectrum of $T_{1}$ -spaced samples of $w (t)$ . Justify your answer by drawing the magnitude spectra of $x_{1}$ , $x_{2}$ , and $x_{3}$ between $- f$ and $f$ .

The triangularly-shaped magnitude spectrum of the real message signal w(t) is shown in the top. The bottom shows the receiver structure in Exercise 6-4. — The triangularly-shaped magnitude spectrum of the real message signal $w (t)$ is shown in the top. The bottom shows the receiver structure in Exercise [link] .

The signal $x (t)$ with magnitude spectrum shown in the top part of [link] is input to the transmission system in the lower part of the Figure.The lowpass filter with output $v_{4} (k T_{s})$ has a cutoff frequency of 200 Hz, a passband gain of 1 and a stopband gain of zero.Specify all frequencies between zero and 2000 Hz at which the magnitudespectra of $| V_{1} (f) |$ , $| V_{2} (f) |$ , $| V_{3} (f) |$ , and $| V_{4} (f) |$ have nonzero values.

The signal x(t) with spectrum X(f) is input into this communication system. Exercise 6-5 asks for the absolute bandwidth of the signal at each point as it moves through the system. — The signal $x (t)$ with spectrum $X (f)$ is input into this communication system. Exercise [link] asks for the absolute bandwidth of the signal at each point as it moves through the system.

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Source: OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3

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