Problem Sheet: Communication Channels Communication Systems

Transcription

1 Problem Shee: Communicaion Channels Communicaion Sysems Professor A. Manikas Chair of Communicaions and Array Processing Deparmen of Elecrical & Elecronic Engineering Imperial College London v.11 Communicaion Channels 1. If one binary source and wo binary channels are conneced in cascade as shown below Binary Source Channel No.1 Channel No.2 where boh channels have he following forward ransiion probabiliy diagram find he bi-error-rae p e a he oupu of he second channel. 10% 2. A digial communicaion sysem, operaing a 100 bis/sec in he presence of addiive whie Gaussian noise of power specral densiy PSD n (f) = N0 2, is represened in he energy uilizaion effi ciency (EUE) - bandwidh uilizaion effi ciency (BUE) plane, as follows: Wha is he capaciy C of he channel in bis/sec? 20% 1

2 E303: Comm Sysems (Communicaion Channels) 2 3. A digial communicaion sysem having an energy uilisaion effi ciency (EUE) equal o 30 operaes in he presence of addiive whie Gaussian noise of double-sided power specral densiy PSD n (f)= W/Hz. If he channel capaciy C is 16 kbis/s and he channel bandwidh B is 4 khz, esimae (a) he bi rae r b 10% (b) he noise power a he channel oupu 10% 4. A discree channel is modelled as follows: Esimae: (a) The probabiliy of error a he oupu of he channel 5% (b) The amoun of informaion delivered a he oupu of he channel 10% 5. Consider a binary Communicaion Sysem ha uses he following wo equally probable energy signals: { } 0 s 0 () = 2Λ 10µs { } 1 s 1 () = 2Λ 10µs The channel is assumed addiive whie Gaussian noise of double-sided power specral densiy PSD n (f) = 10 6 W/Hz. Find: (a) he bandwih B of he channel; 5% (b) he channel symbol rae r cs (baud rae) & daa bi rae; 5% (c) he Energy Uilisaion Effi ciency (EUE); 10% (d) he channel capaciy C in bis/sec. 10% 6. Consider a binary Communicaion Sysem ha operaes wih a bi rae 100kbis/sec and uses he following wo equally probable energy signals: ( { } { }) 0 s 0 () = 3 Λ + rec 5µs 10µs ( { } { }) 1 s 1 () = 3 Λ + rec 5µs 10µs The channel is assumed addiive whie Gaussian noise of double-sided power specral densiy PSD n (f) = W/Hz. Find: (a) he bandwih B of he channel; 5%

3 E303: Comm Sysems (Communicaion Channels) 3 (b) he channel symbol rae r cs (baud rae); 5% (c) he Energy Uilisaion Effi ciency (EUE); 20% (d) he channel capaciy C in bis/sec. 15% 7. Consider a binary digial communicaion sysem { in} which a binary sequence is ransmied { } as a signal s() wih a one being sen as 6Λ and a zero being sen as 6Λ. T cs/2 T cs/2 The source a he inpu o he sysem provides a binary sequence of ones and zeros, wih he number of ones being wice he number of zeros. The ransmied signal is corruped by channel noise n() of bandwidh B and has an ampliude probabiliy densiy funcion described by he following expression: pdf n (n) = 1 6.rec { n 6 Find a bound on he raio C/B 20% where C denoes he capaciy of he channel in bis/s. 8. Consider a binary digial communicaion sysem in which he ransmied signal is corruped by channel noise of bandwidh B having an ampliude probabiliy densiy funcion described by he following expression: pdf n (n) = 1 { n } 6.rec 6 If he power of he received signal is 12W hen (a) find he enropy power of he noise; 10% (b) find an upper and a lower bound on he raio C/B where C denoes he capaciy of he communicaion channel. 10% 9. A discree channel is modelled as follows: } Esimae: (a) The probabiliy of error a he oupu of he channel 5% (b) The amoun of informaion delivered a he oupu of he channel 15%

4 E303: Comm Sysems (Communicaion Channels) A discree channel is modelled as follows: Esimae: (a) The probabiliy of error a he oupu of he channel 5% (b) The amoun of informaion delivered a he oupu of he channel 15% 11. A signal g() bandlimied o 4kHz is sampled a he Nyquis rae and is fed hrough a 2-level quanizer. A Huffman encoder is used o encode riples of successive oupu quanizaion levels as follows: symbols probs Huffman m 1 m 1 m 1 27/64 1 m 1 m 1 m 2 9/ m 1 m 2 m 1 9/ m 2 m 1 m 1 9/ m 1 m 2 m 2 3/ m 2 m 1 m 2 3/ m 2 m 2 m 1 3/ m 2 m 2 m 2 1/ while he binary sequence a he oupu of he Huffman encoder is fed o a Binary on-off Keyed Communicaion Sysem which employs he following wo energy signals of duraion T cs s 0 () = 0 ( ) 3 s 1 () = 8 Λ 0.5 T cs The ransmied signals are corruped by addiive whie Gaussian channel noise having a double-sided power specral densiy of 10 3 W/Hz. The figure below shows a modelling of he whole sysem where he oupu of he Huffman encoder is modelled as he oupu of a binary discree informaion source (X, p) wih X = {x 1 = 1, x 2 = 0}, p = [Pr(x 1 ), Pr(x 2 )] T while he binary on-off Keyed sysem is modelled as a discree channel as shown below.

5 E303: Comm Sysems (Communicaion Channels) 5 (a) Find he enropy of he informaion source (X, p), he informaion rae and he bi daa rae (symbol rae) a he channel inpu. 15% (b) Esimae he bi-error probabiliy of he sysem. 10% (c) Esimae he energy uilizaion effi ciency (EUE) and bandwidh uilizaion effi ciency (BUE) using he bi daa rae as well as he informaion rae. 15% (d) Represen he communicaion sysem, as a poin on he (EUE,BUE) parameer plane. In his plane show also he locus of he sysem properly labelled. 10% (e) Is he sysem a realizable communicaion sysem? 5% (f) Wha is he signal-o-noise raio, SNR in, a he receiver s inpu? 5% 12. A signal g() having he pdf shown in Figure 1 is bandlimied o 4 khz. The signal is sampled a he Nyquis rae and fed hrough a 2-level quanizer. The ransfer funcion of he quanizer is shown in Figure 2. Figure-1 Figure-2 A Huffman encoder is used o encode riples of successive oupu quanizaion levels while he binary sequence a he oupu of he Huffman encoder is fed o a Binary on-off Keyed Communicaion Sysem which employs he following wo energy signals ( s 1 () = 0; s 2 () = 0.5 cos 2π 5 ) ; wih 0 < < T cs T cs The whole sysem is modelled as follows where he binary informaion source represens he sysem up o he oupu of he Huffman encoder. The discree channel models he binary on-off keyed Transmier/Receiver (wih

6 E303: Comm Sysems (Communicaion Channels) 6 x 1 = 1 and x 2 = 0) and he addiive whie Gaussian noisy channel wih noise having a double-sided power specral densiy of 10 3 W/Hz. (a) Esimae he bi-error probabiliy of he sysem. 5% (b) Find he informaion rae and he bi daa rae (symbol rae) a he channel inpu. 10% (c) Esimae he daa poin (EUE,BUE), where EUE denoes he energy uilizaion effi - ciency and BUE represens he bandwidh uilizaion effi ciency of he sysem. 10% (d) Esimae he informaion poin (EUE,BUE), where EUE denoes he informaion energy uilizaion effi ciency and BUE represens he informaion bandwidh uilizaion effi ciency of he sysem. 15% (e) Is he sysem a realizable communicaion sysem? 5% (f) Wha is he signal-o-noise raio SNR, a he receiver s inpu? 5%. END Imperial College London Coursework