Consider the simplest complex constellations of 8-PSK and 16-PSK which sends symbols along both the I...

  

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Consider the simplest complex constellations of 8-PSK and 16-PSK which sends symbols along both the I and Q components, as shown in Fig. 1. Let E denote the average transmit power, and 2 the variance of each of the real and imaginary parts of the additive white Gaussian noise at a receiver. 110 0110 0111 4 0010 . 0101 0011 . . 0001 101 100 001 0100 . JE Admin 22 1100 000 41- 1101 . 111 011 010 1201 . 1110 1011 1010 . 27 1424 " 64 43 " 13 84 " " 0000 1000 . a) Derive the error probability of each of the constellations in a single-input-single-output (SISO) Rayleigh fading channel. Give the results in terms of the Q-function. For analytical tractability, we only take into account the errors between adjacent constellation points. (10 marks) b) Given the signal-to-noise ratio (SNR), i.e., . find the rates of error-free uncoded transmissions using the two constellations. (10 marks) c) Identify the SNR condition under which one constellation can outperform the other. (5 marks) Consider the simplest complex constellations of 8-PSK and 16-PSK which sends symbols along both the I and Q components, as shown in Fig. 1. Let E denote the average transmit power, and 2 the variance of each of the real and imaginary parts of the additive white Gaussian noise at a receiver. 110 0110 0111 4 0010 . 0101 0011 . . 0001 101 100 001 0100 . JE Admin 22 1100 000 41- 1101 . 111 011 010 1201 . 1110 1011 1010 . 27 1424 " 64 43 " 13 84 " " 0000 1000 . a) Derive the error probability of each of the constellations in a single-input-single-output (SISO) Rayleigh fading channel. Give the results in terms of the Q-function. For analytical tractability, we only take into account the errors between adjacent constellation points. (10 marks) b) Given the signal-to-noise ratio (SNR), i.e., . find the rates of error-free uncoded transmissions using the two constellations. (10 marks) c) Identify the SNR condition under which one constellation can outperform the other. (5 marks)