NAME:

INSTRUCTIONS: Please Read If you need more room, write on the back side of the PREVIOUS page, not the back of the current page!! Please do not use your paper. Open book(s) and notes, including any lab related material. Cell phones oﬀ and out of sight. Be sure what you have written down is correct. YOU COULD LOSE POINTS FOR INCORRECT STATEMENTS. There are 6 problems on 4 total pages, including this page, make sure you have all pages now. The tests must be turned in by the time announced for full credit. KEEP YOUR ANSWERS BRIEF AND TO THE POINT. DON’T WASTE TIME WRITING LONG DISCUSSIONS. SOME Useful Facts • xn = f0 G(nf0 ), where xn are the Fourier Series coeﬃcients of periodic signal x(t), and G(f ) is the Fourier Transform of a single period of x(t). • “Convolution of a signal of width w1 with a signal of width w2 results in a signal of width w1 + w2 .” • From the Fourier Transform table: k=∞ n=∞

w(t) =

k=−∞

δ(t − kT )

⇐⇒

W (f ) = 1/T

n=−∞

δ(f − n/T )

2 1. 10% dB problem: The output signal power of an ideal ampliﬁer is 17 dBm. The input signal power is 2 milliWatts. Find the power gain of the ampliﬁer in dB.

2. 20% For all parts, consider a uniform quantizer, with 32 levels, designed for an amplitude range of [−8, +8] volts (in other words, 8 volts peak amplitude, or (±8 volts)). The input is a stationary random signal. (a) Assume the input signal has amplitudes ranging between ±8 volts. Compute the PEAK signal to quantization noise ratio (SQNR) in dB. (b) Assume for this part that the quantizer input signal has amplitude PDF given in the ﬁgure below. Compute the PEAK SQNR of the output signal in dB. (c) Assume for this part that the signal from part (b) is processed by the simple block diagram shown below to form the signal Z(t). Z(t) is input to the quantizer. Determine the maximum and minimum amplitudes for Z(t) in volts. Compute the PEAK SQNR of the quantizer output in dB.

PDF of Amplitude 0.5 1 2 3 4 Amplitude (volts)

signal from part (b)

+ −3 volts voltage gain = 2

Z(t)

3 3. 25% Consider a binary, Antipodal PAM baseband communications system, with bit time Tb . The channel includes additive, white, Gaussian noise, with Sn (f ) = β/2 Watts/Hz. Note that there are 2 systems, System A and System B, and a set of 6 diﬀerent waveforms, z1 (t) · · · z6 (t), all shown in the ﬁgures below. (a) Consider SYSTEM B. Assume s(t) = z1 (t). Circle all waveforms for impulse response h(t) can be equal to such that the signal power to noise power ratio (S/N ) at the random variable YB is the theoretical maximum possible. YOUR ANSWER (circle all that apply): z1 (t), z2 (t), z3 (t), z4 (t), z5 (t), z6 (t), None, ALL (b) Consider SYSTEM A. Assume s(t) = z1 (t). Circle all waveforms for the template waveform w(t) can be equal to such that the signal power to noise power ratio (S/N ) at the random variable YA is the theoretical maximum possible. YOUR ANSWER (circle all that apply): z1 (t), z2 (t), z3 (t), z4 (t), z5 (t), z6 (t), None, ALL (c) For SYSTEM A: ASSUME s(t) = z1 (t). Compute the theoretical maximum S/N at the output YA . For full credit, your answer should be in terms of the signal amplitude(s) and given noise spectral density. (d) Assume that the transmitter has a peak amplitude limitation. Which of the 6 waveforms would give the best performance if used as a transmitted waveform s(t), assuming a matched ﬁlter receiver is used? YOUR ANSWER (circle all that apply): z1 (t), z2 (t), z3 (t), z4 (t), z5 (t), z6 (t), None, ALL SYSTEM A s(t) + n(t)

X

Tb dt 0

SYSTEM B t=T Y A b s(t) + n(t) a a/2 z (t) 2 b b/2 Tb/4 3Tb /4 Tb t Tb/4 3Tb /4 z (t) 6 Tb t impulse response h(t) t=T b Y B

w(t) z (t) 1

a a/2

z (t) 3

Tb/4

3Tb /4

Tb

t

z (t) 4 3Tb/4 T /4 Tb b t −a/2 −a

a a/2

z (t) 5

a a/2

Tb/4

3Tb /4

Tb

t

Tb/4

3Tb /4

Tb

t

4 4. 10% A 64 QAM system is used to transmit data at a bit rate of...