Digital Comminication

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 𝑠1 (𝑡), 𝑠1 (𝑡), 𝑠1 (𝑡) and 𝑠1 (𝑡) are 𝑝 = [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 𝑠𝑚 (𝑡) = 𝐴𝑚 𝑝(𝑡), For each case find average signal energy (assume equiprobable signals). 𝑝(𝑡) a. 𝑀 = 2 , 𝐴 = {±1} 𝑝(𝑡) b. 𝑀 = 2 , 𝐴 = {±1} c. 𝑀 = 4 , 𝐴 = {±1, ±3} d. 𝑀 = 4 , 𝐴 = {±1, ±3} 𝟒 𝟐 4- Suppose 𝐯⃑ = ( 𝟎 ) and 𝐮 ⃑ =( 𝟏 ) −𝟑 −𝟐 a. Find ⟨v ⃑ ,u ⃑⟩ b. Find ⟨u ⃑ ,u ⃑⟩ c. Find ⟨v ⃑ ,v ⃑⟩ d. Find ||v ⃑ || e. Find ||u ⃑ || 𝑝(𝑡) 𝑝(𝑡) 5- Consider the four vectors below: −𝟐 −𝟏 𝟏 𝟐 𝐯⃑ (𝟏) = (−𝟔) , 𝐯⃑ (𝟐) = (−𝟑) , 𝐯⃑ (𝟑) = ( 𝟑 ) , 𝐯⃑ (𝟒) = ( 𝟔 ) 𝟐 𝟏 −𝟏 −𝟐 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 𝑠1 (𝑡) and 𝑠2 (𝑡) shown below. a. Find the energy of each signal. b. Find their inner product ⟨𝑠1 (𝑡), 𝑠2 (𝑡)⟩. 8- Consider the four waveforms as shown below: Consider the following orthonormal functions: a) ∅1 (𝑡) = b) ∅2 (𝑡) = 𝑠1 (𝑡) √𝐸𝑠1 𝑠2 (𝑡) √𝐸𝑠2 = = c) ∅3 (𝑡) = 𝑠3 (𝑡) − 𝑠1 (𝑡) = 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 𝑆𝑚 (𝑡) = 𝐴𝑚 𝑝(𝑡) 𝐴𝑚 = 2𝑚 − 1 − 𝑀 Basis: ∅(𝑡) = 𝑝(𝑡) √𝐸𝑝 , 𝑆𝑚 (𝑡) = 𝑔(𝑡) cos(2𝜋𝑓𝑐 𝑡 + 𝜃𝑚 )  𝑆𝑚 (𝑡) = 𝐴𝑚 √𝐸𝑝 ∅(𝑡) M-QAM 𝑆𝑚 (𝑡) = 𝐴𝑚 (𝐼) Basis: ∅1 (𝑡) = √ 𝐸𝑔 𝑔(𝑡) cos(2𝜋𝑓𝑐 𝑡) − 𝐴𝑚 𝑔(𝑡) cos(2𝜋𝑓𝑐 𝑡) 𝐸 𝑆𝑚 (𝑡) = √ 𝑔 𝐴𝑚 2 (𝐼) 2 Basis: ∅1 (𝑡) = √ 𝐸𝑔 𝑔(𝑡) cos(2𝜋𝑓𝑐 𝑡) ∅2 (𝑡) = −√ 2 𝐸𝑔 𝐸 𝐸 2 2 𝑔(𝑡) sin(2𝜋𝑓𝑐 𝑡) 𝑆𝑚 (𝑡) = √ 𝑔 cos(𝜃𝑚 ) ∅1 (𝑡) + √ 𝑔 sin(𝜃𝑚 ) ∅2 (𝑡) (𝑄) 𝑔(𝑡) sin(2𝜋𝑓𝑐 𝑡) 𝐴𝑚 (𝐼) = 𝐴𝑚 (𝑄) = 2𝑚 − 1 − √𝑀 , 𝑚 = 1,2, . . , √𝑀 2 2𝜋 𝜃𝑚 = 𝑀 (𝑚 − 1) , 𝑚 = 1,2, . . , 𝑀 𝑚 = 1,2, . . , 𝑀 ∅2 (𝑡) = −√ 2 𝐸𝑔 𝐸 ∅1 (𝑡) + √ 𝑔 2 𝑔(𝑡) sin(2𝜋𝑓𝑐 𝑡) 𝐴𝑚 (𝑄) ∅2 (𝑡) M-FSK 𝑆𝑚 (𝑡) = 𝐴 cos(2𝜋𝑓𝑚 𝑡) 𝑓𝑚 = 𝑚∆𝑓 , 𝑚 = 1,2, . . , 𝑀 M-ASK 𝑆𝑚 (𝑡) = 𝐴𝑚 𝑔(𝑡)cos(2𝜋𝑓𝑐 𝑡) 𝐴𝑚 = 2𝑚 − 1 − 𝑀 , 𝑚 = 1,2, . . , 𝑀 1- Find the Gray code for the following binary block length (b): a. 𝑏 = 1 b. 𝑏 = 2 c. 𝑏 = 3 1|Page 2- Draw the constellation diagrams for: 2-PAM 4-PAM ∅2 (𝑡) ∅(𝑡) Index (m) Binary Block (b) Amplitude (𝐴𝑚 ) ∅1 (𝑡) Vector 𝑆 (𝑚) Index (m) Binary Block (b) Amplitude (𝐴𝑚 ) BPSK QPSK ∅2 (𝑡) ∅2 (𝑡) Vector 𝑆 (𝑚) ∅1 (𝑡) Index (m) Binary Block (b) Phase (𝜃𝑚 ) ∅1 (𝑡) Vector 𝑆 (𝑚) Index (m) Binary Block (b) Phase (𝜃𝑚 ) Vector 𝑆 (𝑚) 8-PSK ∅2 (𝑡) ∅1 (𝑡) Index (m) Binary Block (b) Phase (𝜃𝑚 ) Vector 𝑆 (𝑚) Index (m) Binary Block (b) Phase (𝜃𝑚 ) Vector 𝑆 (𝑚) 2|Page 4-QAM ∅2 (𝑡) Index (m) Binary Block (b) Amplitude (𝐴𝑚 (𝐼) ) (𝐴𝑚 (𝑄) ) ∅1 (𝑡) Index (m) Binary Block (b) Amplitude (𝐼) (𝐴 𝑚 ) (𝐴 𝑚 (𝑄) Vector 𝑆 (𝑚) ) 16-QAM ∅2 (𝑡) Index (m) Binary Block (b) Amplitude (𝐴𝑚 (𝐼) ) (𝐴𝑚 (𝑄) ) ∅1 (𝑡) Index (m) Binary Block (b) Amplitude (𝐼) (𝐴 𝑚 ) (𝐴 𝑚 (𝑄) ) Vector 𝑆 (𝑚) Index (m) Binary Block (b) Amplitude (𝐼) (𝐴 𝑚 ) (𝐴 𝑚 (𝑄) ) Vector 𝑆 (𝑚) 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 𝒑(𝒕) with amplitude 𝑨 and duration 𝑻. a. Draw the transmitted signal 𝒔(𝒕), assume PAM (M=2) with 𝑻𝒃 = 𝑻. 𝑻 b. Draw the transmitted signal 𝒔(𝒕), assume PAM (M=2) with 𝟐𝒃 = 𝑻. c. Draw the transmitted signal 𝒔(𝒕), assume 4-PAM (M=4) with 𝑻𝒔 = 𝑻. +3 +1 -1 -3 5- You want to transmit the binary sequence (10010011) using ASK signaling with: a. M=2, Carrier frequency 𝒇𝒄 = 𝟏⁄𝑻 . 𝒔 5|Page b. M=4, Carrier frequency 𝒇𝒄 = 𝟏⁄𝑻 . 𝒔 +3 +1 -1 -3 6- You want to transmit the binary sequence (10010011) using M-PSK signaling with: a. 𝑴 = 𝟐 , Carrier frequency 𝒇𝒄 = 𝟏⁄𝑻 . 𝒔 b. 𝑴 = 𝟐 , Carrier frequency 𝒇𝒄 = 𝟐⁄𝑻 . 𝒔 6|Page c. 𝑴 = 𝟒 , Carrier frequency 𝒇𝒄 = 𝟏⁄𝑻 . 𝒔 7- You want to transmit the binary sequence (10010011) using M-FSK signaling with: a. 𝑴 = 𝟐 , ∆𝒇 = 𝟏⁄𝑻 . 𝒔 b. 𝑴 = 𝟒 , ∆𝒇 = 𝟏⁄𝑻 . 𝒔 7|Page EE352- Digital Communication Exercise [Chapter (10)] Name: ID: Date: Problem 1. Determine the autocorrelation function 𝑅𝑥 (𝜏) and the power 𝑃𝑥 of a low-pass random process with a white noise PSD 𝑆𝑥 (𝑓) = 𝑁⁄2 as shown in figure below. 𝑆𝑥 (𝑓) 𝑁⁄ 2 𝑓 −𝐵 Page 1 of 4 𝐵 Problem 2. Consider a random process 𝑥 (𝑡) = 𝐴 cos(2𝜋𝑓𝑐 𝑡 + 𝜃) Where 𝐴 and 𝑓𝑐 are constants and 𝜃 is an RV uniformly distributed over (0 , 2𝜋). 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 𝑆𝑥 (𝑓) = 4 𝛿(𝑓 − 10) and the transfer function of the system is 𝐻(𝑓) = the output signal 𝑆𝑦 (𝑓). 𝑥(𝑡) 𝑦(𝑡) 𝐻(𝑓) Page 3 of 4 1 1+𝑗 3𝑓 . Find the PSD of Problem 4. FIND THE 90% BANDWIDTH for the signal whose power spectral density is as given below: 𝑆𝑥 (𝑓) 2 −20 Page 4 of 4 20 𝑓 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. 𝑚 ̂ (𝑟(𝑡)) = 𝑚ℓ ∶ ℓ = argmax Pr{𝑚 = 𝑚𝑖 |𝑟(𝑡)} 𝑖 Problem 2. Assume M=4. Draw a block-diagram of a maximum likelihood (ML) receiver that uses the following decision rule. 𝑚 ̂ (𝑟(𝑡)) = 𝑚ℓ ∶ Page 1 of 5 ℓ = argmax Pr{𝑟 (𝑡) | 𝑚𝑖 𝑖𝑠 𝑠𝑒𝑛𝑡} 𝑖 Problem 3. Assume M=4. Draw a block-diagram of a minimum Euclidean distance (MED) receiver that uses the following decision rule. 𝑇𝑠 𝑚 ̂ (𝑟(𝑡)) = 𝑚ℓ ∶ ℓ = argmax ∫ 𝑟(𝑡) 𝑧𝑖 (𝑡)𝑑𝑡 − 𝑖 𝐸𝑖⁄ 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. 𝑚 ̂ (𝑟(𝑡)) = 𝑚ℓ ∶ ℓ = argmax 𝑟(𝑡) ∗ 𝑧𝑖 (𝑇𝑠 − 𝑡)|𝑡=(𝑛+1)𝑇 − 𝐸𝑖⁄ 𝑖 𝑠 2 𝑟(𝑡) 𝑧𝑖 (𝑡 − 𝑛𝑇𝑠 )𝑑𝑡 − 𝐸𝑖⁄ 2 (𝑛+1)𝑇𝑠 𝑚 ̂ (𝑟(𝑡)) = 𝑚ℓ ∶ Page 2 of 5 ℓ = argmax ∫ 𝑖 𝑛𝑇𝑠 Problem 5. Assume a Minimum Euclidean Distance reciver using correlator as shown below, For 𝑀 = 2. The received signal alternatives are such that 𝑧0 (𝑡) = − 𝑧1 (𝑡) and, 𝑧1 (𝑡) 𝐵 𝑇𝑏 2 𝑇𝑏 𝑡 Furthermore, assume a noisefree situation and that 𝑟(𝑡) = 𝑧1 (𝑡). Calculate the two decision variables 𝜉0 , 𝜉1 , and also the decision 𝑚 ̂ . Is the decision correct? Page 3 of 5 Problem 6. Assume a Minimum Euclidean Distance reciver using correlator as shown below, For 𝑀 = 2. The received signal alternatives are such that 𝑧0 (𝑡) = − 𝑧1 (𝑡) and, 𝑧1 (𝑡) 𝐵 𝑇𝑏 2 𝑇𝑏 𝑡 Furthermore, assume 𝑟(𝑡) = 𝑧1 (𝑡) + 𝑁(𝑡), where 𝑁(𝑡) is AWGN eith power spectral density 𝑁0⁄ V 2⁄ 2 [ Hz]. Due to the noise 𝑁(𝑡), the decision variables 𝜉0 , 𝜉1 also contain a noise component 𝑁 (𝑤 and −𝑤 respectively). The noise 𝑤 has zero mean and variance 𝜎𝑤2 = 20 𝐸0 (where 𝐸0 is the energy of signal 𝑧0 (𝑡)). Determine the probability of a “miss” in terms of the 𝐐()-function. 𝐸 Calculate 𝑃𝑀 , if 0⁄𝑁 is 12.55 [dB]. 0 Page 4 of 5 Problem 7. a) Assume 𝑃0 = 𝑃1 and a Minimum Euclidean Distance reciver using correlator. How large 𝐸𝑏 ⁄𝑁 in dB, is needed to obtain 𝑃𝑏 = 10−5 if the received signal alternatives 𝑧0 (𝑡) 𝑎𝑛𝑑 𝑧1 (𝑡) 0 are antipodal signals. b) Repeat the calculation in (a) but assume orthogonal signals. Page 5 of 5

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