EE352- Digital Communication Exercise [Chapter (9)] Name: ID: Date: 1- Consider 16 possible message signals transmitted using PAM. The signal interval is 0.1 msec. Find: a. Symbol rate (baud rate) b. Bit rate 2- Consider 4 message signals as shown below: a. Find energy of each signal. b. If the probability of messages 1 ( ), 1 ( ), 1 ( ) and 1 ( ) are = [0.1 0.3 0.5 0.1]. Find the average signal energy per symbol. c. Find average energy per bit. d. If equiprobable signals. Find the average signal energy per symbol. 3- For M message signals we use PAM modulator ( ) = ( ), For each case find average signal energy (assume equiprobable signals). ( ) a. = 2 , = {±1} ( ) b. = 2 , = {±1} c. = 4 , = {±1, ±3} d. = 4 , = {±1, ±3} 4- Suppose ⃑ = ( ) and ⃑ =( ) − − a. Find 〈v ⃑ ,u ⃑〉 b. Find 〈u ⃑ ,u ⃑〉 c. Find 〈v ⃑ ,v ⃑〉 d. Find ||v ⃑ || e. Find ||u ⃑ || ( ) ( ) 5- Consider the four vectors below: − − ⃑ ( ) = (− ) , ⃑ ( ) = (− ) , ⃑ ( ) = ( ) , ⃑ ( ) = ( ) − − Represent these vectors in one-dimensional representation. −1 2 5 5 0 −1 6- Consider three vectors: v ⃑ (1) = ( ) , ⃑v (2) = ( ) , ⃑v (3) = ( ). Let : −5 0 1 1 −2 −5 1 −1 1 1 1 1 e⃑(1) = 2 ( ) , e⃑(2) = 2 ( ) −1 −1 −1 1 a. Show that e⃑(1) and e⃑(2) are orthonormal. b. Calculate the following inner products: 〈v ⃑ (1) , e⃑(1) 〉 = 〈v ⃑ (1) , e⃑(2) 〉 = 〈v ⃑ (2) , e⃑(1) 〉 = 〈v ⃑ (2) , e⃑(2) 〉 = 〈v ⃑ (3) , e⃑(1) 〉 = 〈v ⃑ (3) , e⃑(2) 〉 = c. Suppose we use e⃑(1) and e⃑(2) as the new axes. Find the corresponding vectors c (1) , c (2) and c (3) that represent v ⃑ (1) , v ⃑ (2) and v ⃑ (3) in the new coordinate system defined by e⃑(1) and e⃑(2) . 7- Consider the two signals 1 ( ) and 2 ( ) shown below. a. Find the energy of each signal. b. Find their inner product 〈 1 ( ), 2 ( )〉. 8- Consider the four waveforms as shown below: Consider the following orthonormal functions: a) ∅1 ( ) = b) ∅2 ( ) = 1 ( ) √ 1 2 ( ) √ 2 = = c) ∅3 ( ) = 3 ( ) − 1 ( ) = Write the four previous signals waveforms in vector form. EE352- Digital Communication Exercise [Chapter (9) – Part B] Name: ID: Date: Notes: M-PAM M-PSK ( ) = ( ) = 2 − 1 − Basis: ∅( ) = ( ) √ , ( ) = ( ) cos(2 + ) ( ) = √ ∅( ) M-QAM ( ) = ( ) Basis: ∅1 ( ) = √ ( ) cos(2 ) − ( ) cos(2 ) ( ) = √ 2 ( ) 2 Basis: ∅1 ( ) = √ ( ) cos(2 ) ∅2 ( ) = −√ 2 2 2 ( ) sin(2 ) ( ) = √ cos( ) ∅1 ( ) + √ sin( ) ∅2 ( ) ( ) ( ) sin(2 ) ( ) = ( ) = 2 − 1 − √ , = 1,2, . . , √ 2 2 = ( − 1) , = 1,2, . . , = 1,2, . . , ∅2 ( ) = −√ 2 ∅1 ( ) + √ 2 ( ) sin(2 ) ( ) ∅2 ( ) M-FSK ( ) = cos(2 ) = ∆ , = 1,2, . . , M-ASK ( ) = ( )cos(2 ) = 2 − 1 − , = 1,2, . . , 1- Find the Gray code for the following binary block length (b): a. = 1 b. = 2 c. = 3 1|Page 2- Draw the constellation diagrams for: 2-PAM 4-PAM ∅2 ( ) ∅( ) Index (m) Binary Block (b) Amplitude ( ) ∅1 ( ) Vector ( ) Index (m) Binary Block (b) Amplitude ( ) BPSK QPSK ∅2 ( ) ∅2 ( ) Vector ( ) ∅1 ( ) Index (m) Binary Block (b) Phase ( ) ∅1 ( ) Vector ( ) Index (m) Binary Block (b) Phase ( ) Vector ( ) 8-PSK ∅2 ( ) ∅1 ( ) Index (m) Binary Block (b) Phase ( ) Vector ( ) Index (m) Binary Block (b) Phase ( ) Vector ( ) 2|Page 4-QAM ∅2 ( ) Index (m) Binary Block (b) Amplitude ( ( ) ) ( ( ) ) ∅1 ( ) Index (m) Binary Block (b) Amplitude ( ) ( ) ( ( ) Vector ( ) ) 16-QAM ∅2 ( ) Index (m) Binary Block (b) Amplitude ( ( ) ) ( ( ) ) ∅1 ( ) Index (m) Binary Block (b) Amplitude ( ) ( ) ( ( ) ) Vector ( ) Index (m) Binary Block (b) Amplitude ( ) ( ) ( ( ) ) Vector ( ) 3|Page 3- Draw the transmitted signal for an input binary sequence (10001001) assuming: a. Amplitude Shift Keying (ASK) b. Frequency Shift Keying (FSK) c. Binary Phase Shift Keying (BPSK) a. ASK b. FSK c. BPSK 4|Page 4- You want to transmit the binary sequence (10010011) using a rectangular pulse ( ) with amplitude and duration . a. Draw the transmitted signal ( ), assume PAM (M=2) with = . b. Draw the transmitted signal ( ), assume PAM (M=2) with = . c. Draw the transmitted signal ( ), assume 4-PAM (M=4) with = . +3 +1 -1 -3 5- You want to transmit the binary sequence (10010011) using ASK signaling with: a. M=2, Carrier frequency = ⁄ . 5|Page b. M=4, Carrier frequency = ⁄ . +3 +1 -1 -3 6- You want to transmit the binary sequence (10010011) using M-PSK signaling with: a. = , Carrier frequency = ⁄ . b. = , Carrier frequency = ⁄ . 6|Page c. = , Carrier frequency = ⁄ . 7- You want to transmit the binary sequence (10010011) using M-FSK signaling with: a. = , ∆ = ⁄ . b. = , ∆ = ⁄ . 7|Page EE352- Digital Communication Exercise [Chapter (10)] Name: ID: Date: Problem 1. Determine the autocorrelation function ( ) and the power of a low-pass random process with a white noise PSD ( ) = ⁄2 as shown in figure below. ( ) ⁄ 2 − Page 1 of 4 Problem 2. Consider a random process ( ) = cos(2 + ) Where and are constants and is an RV uniformly distributed over (0 , 2 ). Determine: a) b) c) d) e) f) Page 2 of 4 Sketch the ensemble of this random process. The mean value. The autocorrelation function. The mean square value. The power spectral density. Is the process wide sense stationary? Problem 3. Consider a linear-time invariant (LTI) system shown below. If the PSD of the input signal given by ( ) = 4 ( − 10) and the transfer function of the system is ( ) = the output signal ( ). ( ) ( ) ( ) Page 3 of 4 1 1+ 3 . Find the PSD of Problem 4. FIND THE 90% BANDWIDTH for the signal whose power spectral density is as given below: ( ) 2 −20 Page 4 of 4 20 Name_____________________ ID __________ EE 352: Digital Communications Exercise [Chapter 11] Problem 1. Assume M=4. Draw a block-diagram of a maximum-a-posteriori probability (MAP) receiver that uses the following decision rule. ̂ ( ( )) = ℓ ∶ ℓ = argmax Pr{ = | ( )} Problem 2. Assume M=4. Draw a block-diagram of a maximum likelihood (ML) receiver that uses the following decision rule. ̂ ( ( )) = ℓ ∶ Page 1 of 5 ℓ = argmax Pr{ ( ) | } Problem 3. Assume M=4. Draw a block-diagram of a minimum Euclidean distance (MED) receiver that uses the following decision rule. ̂ ( ( )) = ℓ ∶ ℓ = argmax ∫ ( ) ( ) − ⁄ 2 0 Problem 4. Assume M=4. Draw a block-diagram of a matched filter (MF) receiver and correlator receiver that use the following decision rule. ̂ ( ( )) = ℓ ∶ ℓ = argmax ( ) ∗ ( − )| =( +1) − ⁄ 2 ( ) ( − ) − ⁄ 2 ( +1) ̂ ( ( )) = ℓ ∶ Page 2 of 5 ℓ = argmax ∫ Problem 5. Assume a Minimum Euclidean Distance reciver using correlator as shown below, For = 2. The received signal alternatives are such that 0 ( ) = − 1 ( ) and, 1 ( ) 2 Furthermore, assume a noisefree situation and that ( ) = 1 ( ). Calculate the two decision variables 0 , 1 , and also the decision ̂ . Is the decision correct? Page 3 of 5 Problem 6. Assume a Minimum Euclidean Distance reciver using correlator as shown below, For = 2. The received signal alternatives are such that 0 ( ) = − 1 ( ) and, 1 ( ) 2 Furthermore, assume ( ) = 1 ( ) + ( ), where ( ) is AWGN eith power spectral density 0⁄ V 2⁄ 2 [ Hz]. Due to the noise ( ), the decision variables 0 , 1 also contain a noise component ( and − respectively). The noise has zero mean and variance 2 = 20 0 (where 0 is the energy of signal 0 ( )). Determine the probability of a “miss” in terms of the ()-function. Calculate , if 0⁄ is 12.55 [dB]. 0 Page 4 of 5 Problem 7. a) Assume 0 = 1 and a Minimum Euclidean Distance reciver using correlator. How large ⁄ in dB, is needed to obtain = 10−5 if the received signal alternatives 0 ( ) 1 ( ) 0 are antipodal signals. b) Repeat the calculation in (a) but assume orthogonal signals. Page 5 of 5

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