MBWC Lab

Discrete Multi-tone(LAB 5)

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MBWC Lab

Requirement

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• Generate the channel with different length (𝐡 is known)

𝐡 = [ℎ0, … , ℎ5] (delay with 5 time slots)

𝐡 = [ℎ0, … , ℎ9] (delay with 9 time slots)

• Generate 64 QPSK symbols to form a signal vector in frequency domain

• Redo the same process with 1000 realizations

• Show the QPSK symbol error rate in different SNR

MBWC Lab

Channel Delay ≤ The Length of CP (1/2)

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• The received signal (𝐇 is a circular matrix) in time domain

𝐱 = 𝐇𝐬 + 𝐰

• The signal from time domain to frequency domain (𝐐 is an FFT matrix)

𝐒 = 𝐐𝐬 𝐗 = 𝐐𝐱

• The circular matrix 𝐇 (𝚲 is a diagonal matrix)

𝐇 = 𝐐H𝚲𝐐

• The received signal in frequency domain

𝐐𝐱 = 𝐐 𝐇𝐬 + 𝐰

→ 𝐐𝐱 = 𝐐 𝐐H𝚲𝐐𝐐H𝐒 + 𝐰 = 𝐐𝐐H 𝚲 𝐐𝐐H 𝐒 + 𝐐𝐰

→ 𝐗 = 𝚲𝐒 + 𝐐𝐰

MBWC Lab

Channel Delay ≤ The Length of CP (2/2)

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• Generate 64 QPSK symbols to form a signal vector, 𝐒 ∈ 𝐂64×1, in frequencydomain

• Generate the circular matrix, 𝐇 ∈ 𝐂64×64, and derive the correspondingdiagonal matrix, 𝚲 (𝐇 = 𝐐H𝚲𝐐)

• Derive the received signal in time domain

𝐱 = 𝐇𝐬 + 𝐰 = 𝐐H𝚲𝐐𝐐H𝐒 + 𝐰

• Decode the signal vector 𝐒 ∈ 𝐂64×1(in frequency domain as follow)

𝐒 = 𝚲−1𝐐 𝐱 = 𝚲−1𝐐 𝐇𝐬 + 𝐰 = 𝚲−1𝐐 𝐐H𝚲𝐐𝐐H𝐒 + 𝐰

→ 𝐒 = 𝐒 + 𝚲−1𝐐 𝐰

MBWC Lab

Channel Delay ≥ The Length of CP

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The ISI exists here !!

MBWC Lab

The Symbol Error Rate

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Channel Delay ≤ The Length of CP Channel Delay ≥ The Length of CP