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ECE 459 Fall 2000
Handout # 3 September 12, 2000
HOMEWORK ASSIGNMENT 2
Reading: Lecture notes (lectures 4-7), Proakis (Chapter 4), papers referenced in lecture notes.
Due Date: Tuesday, September 26, 2000 (in class)
- (10 pts) Simplex Signal Set. Consider a set of M orthogonal signal waveforms {sm(t)}M m^ =0−^1 that have energy E. Define a new set of M waveforms as
s′ m(t) = sm(t) −
M
M∑ − 1
`=
s`(t) , m = 0, 1 ,... , M − 1.
Show that the M signal waveforms have equal energy, given by
1 − M^1
E, and are equally cor- related, with correlation coefficients ρkm = − (^) M^1 − 1 and distances dkm =
2 E.
- (20 pts) Signal Constellation Optimization. Consider the QAM signal constellation shown in Figure 1.
r 1
r 2
Figure 1: Signal constellation for Problem 2.
(a) (8 pts) Evaluate ζ = d
(^2) min Eb as a function of^ a^ =^
r 2 r 1. (b) (8 pts) Maximize ζ(a) over a ≥ 1 to find the best constellation. (c) (4 pts) Compare the result in (b) with ζ for 8-PSK.
- (10 pts) Phase Trellis for CPM. Determine the number of states required and draw the state trellis for binary CPM with
(a) (4 pts) h = 13 , full response, and q(t) = (^) Tt 11 {t∈[0,T )} + 1 (^1) {t∈[T,∞)}. (b) (6 pts) h = 12 , partial response, and q(t) = 4 tT 11 {t∈[0, 2 T )} + (^12 11) {t∈[2T,∞)}.
- (15 pts) Alternative derivation of the PSD of linearly modulated signals. Let {Bn} be a zero-mean discrete-time WSS complex random process with ACF RB (k) = E[Bn+kB?n] that represents a sequence of digital symbols. Define the PSD of {Bn} by SB (f ) =
k=−∞ RB^ (k)e −j 2 πf k (^) as we did in class. The PSD of the cyclostationary linearly modulated process:
s(t) =
∑^ ∞
n=−∞
Bng(t − nTs)
was derived in class by averaging the periodic ACF Rs(t + τ, t) over the period Ts, and then evaluating the Fourier transform of the average ACF. An alternative approach is to change the cyclostationary process into a stationary one by adding a random delay ∆ (independent of {Bn}) that is uniformly distributed on [0, Ts] to produce:
s¯(t) =
∑^ ∞
n=−∞
Bng(t − nTs − ∆)
and defining the PSD of s(t) to be the PSD of the stationary process ¯s(t). Show that this method produces the same PSD as the one derived in class, i.e., that
Ss¯(f ) =
SB (f Ts)|G(f )|^2 Ts
- (25 pts) Bandwidth of Digitally Modulated Signals. The PSD of Ss(f ) of a linearly modulated digital signal s(t) is determined by |G(f )|^2 , where G(f ) is the Fourier transform of the pulse shaping waveform g(t). There are various ways to define the bandwidth B of a digitally modulated signal in terms of Ss(f ). - Null-to-Null Bandwidth: The width of the main spectral lobe. - 3 -dB Bandwidth: The width of the interval between the two frequencies at which the PSD is 3 dB below its peak value. - x-% Essential Bandwidth: The width of the (smallest) band of frequencies that contains x-% of the total signal energy.
To compare the bandwidths of modulation schemes that use different constellation sizes, it is convenient to normalize the bandwidth by the bit rate 1/Tb. The normalized bandwidth β = BTb.
(a) (18 pts) Compute the normalized bandwidths based on all three definitions for the following digital modulation schemes:
- QPSK with rectangular signaling pulse:
g(t) =
Ts
pTs (t).
- BPSK with time-domain raised cosine (TDRC) signaling pulse:
g(t) =
3 Ts
[
1 + cos 2 π Ts
t − Ts 2
)]
pTs (t).
