0.5 Sampling with automatic gain control  (Page 6/19)

The message signal $u\left(t\right)$ and additive interferer $w\left(t\right)$ with magnitude spectra shown in [link] are applied to the system in [link] . The analog mixer frequencies are $f_c=1600$ kHz and $f_d=1240$ kHz. The BPF with output $x\left(t\right)$ is assumed ideal, is centered at $f_c$ , has lower cutoff frequency $f_L$ , upper cutoff frequency $f_U$ , and zero phase at $f_c$ . The period of the sampler is $T_s=\frac{1}{71}\times {10}^{-4}$ seconds. The phase $\beta$ of the discrete-time mixer is assumed to be adjusted to the value that maximizes the ratioof signal to interferer noise power in $y\left(k{T}_s\right)$ . The LPF with output $y\left(k{T}_s\right)$ is assumed ideal with cutoff frequency $\gamma$ . The design objective is for the spectrum of $y\left(k{T}_s\right)$ to estimate the spectrum of a sampled $u\left(t\right)$ . You are to select the upper and lower cutoff frequencies of theBPF, the frequency $\alpha$ of the discrete-time mixer, and the cutoff frequency of the LPF in order to meet this objective.

1. Design the desired BPF by specifying its upper $f_U$ and lower $f_L$ cutoff frequencies.
2. Compute the desired discrete-time mixer frequency $\alpha$ .
3. Design the desired LPF by specifying its cutoff frequency $\gamma$ .

Consider the digital receiver in [link] producing $y\left(k{T}_s\right)$ , which is intended to match the input $x\left(t\right)$ sampled every $T_s$ seconds. The absolute bandwidth of $x\left(t\right)$ is $B$ . The carrier frequency $f_c$ is 10 times $B$ . The sample frequency $1/T_s$ is 2.5 times $f_c$ . Note that the sample frequency $1/T_s$ is above the Nyquist frequency of the received signal $r\left(t\right)$ . Determine the maximum cutoff frequency as a function of theinput bandwidth $B$ for the lowpass filter producing $y\left(k{T}_s\right)$ so the design objective of matching samples of $x\left(t\right)$ with a sample frequency of $1/T_s$ is achieved.