2.1 Frequency domain problems (ticket #6364)  (Page 5/6)

Catching speeders with radar

RU Electronics has been contracted to design a Doppler radar system. Radar transmitters emit a signal that bounces off any conducting object.Signal differences between what is sent and the radar return is processed and features of interest extracted. In Doppler systems, the object's speed along the direction of the radar beam is the feature the design must extract. The transmitted signal is a sinsusoid: $x(t)=A\cos (2\pi f_{c}t)$ . The measured return signal equals $B\cos (2\pi ((f_c+\mathrm{\Delta f})t+\varphi ))$ , where the Doppler offset frequency $\mathrm{\Delta f}$ equals $10v$ , where $v$ is the car's velocity coming toward the transmitter.

1. Design a system that uses the transmitted and return signals as inputs and produces $\mathrm{\Delta f}$ .
2. One problem with designs based on overly simplistic design goals is that they are sensitive to unmodeled assumptions. How would you change your design, if at all, so that whether the car is going away or toward the transmitter could be determined?
3. Suppose two objects traveling different speeds provide returns. How would you change your design, if at all, to accomodate multiple returns?

Demodulating an am signal

Let $m(t)$ denote the signal that has been amplitude modulated. $$x(t)=A(1+m(t))\sin (2\pi f_{c}t)$$ Radio stations try to restrict the amplitude of thesignal $m(t)$ so that it is less than one in magnitude. The frequency $f_c$ is very large compared to the frequency content of the signal. What we are concerned about here is nottransmission, but reception.

1. The so-called coherent demodulator simply multiplies the signal $x(t)$ by a sinusoid having the same frequency as thecarrier and lowpass filters the result. Analyze this receiver and show that it works. Assume thelowpass filter is ideal.
2. One issue in coherent reception is the phase of the sinusoid used by the receiver relative to that usedby the transmitter. Assuming that the sinusoid of the receiver has a phase $\phi$ , how does the output depend on $\phi$ ? What is the worst possible value for this phase?
3. The incoherent receiver is more commonly used because of the phase sensitivity problem inherent incoherent reception. Here, the receiver full-wave rectifies the received signal and lowpass filtersthe result (again ideally). Analyze this receiver. Does its output differ from that of the coherentreceiver in a significant way?

Unusual amplitude modulation

We want to send a band-limited signal having the depicted spectrum with amplitude modulation in the usual way. I.B. Different suggests using the square-wave carrier shown below . Well, it is different, but his friends wonder if any technique can demodulate it.

1. Find an expression for $X(f)$ , the Fourier transform of the modulated signal.
2. Sketch the magnitude of $X(f)$ , being careful to label important magnitudes and frequencies.
3. What demodulation technique obviously works?
4. I.B. challenges three of his friends to demodulate $x(t)$ some other way. One friend suggests modulating $x(t)$ with $\cos \left(\frac{\pi t}{2}\right)$ , another wants to try modulating with $\cos (\pi t)$ and the third thinks $\cos \left(\frac{3\pi t}{2}\right)$ will work. Sketch the magnitude of the Fourier transform of the signal each student's approach produces.Which student comes closest to recovering the original signal? Why?