(1) A binary PCM signaling uses on-off signaling to transmit symbols 1 and 0 (symbol 1 is represented by a rectangular pulse of amplitude A and duration Tb; symbol 0 is represented by no pulse transmission for period Tb). The additive noise at the receiver input is white and Gaussian with zero mean and PSD of N₀/2. Find an expression for the average probability of error with matched filtering assuming symbol 1 and 0 occur with equal probability.

(2) An analog signal was sampled, quantized and encoded into a binary PCM wave. The sampling frequency was 10 kHz and the number of representation levels was 256. The PCM signal streamed over a baseband channel. How many digits are used to represent each sample? What is the bit rate? How long does it take to transmit one sample? What is the minimum required transmission bandwidth?

(3) The combination of the transmitter and channel produces a pulse with its amplitude spectrum defined by G(f)H(f) = e^-|f|T_b. Determine the spectrum of the of the ideal receiver filter that would eliminate intersymbol interference. Assume ideal Nyquist channel.

(4) In a single parity-check code, a single parity bit is appended to a block of k message bits (m₀, m₁, m_k-1). This single parity bit, b₁, is chosen so that the codeword satisfies the even parity rule: m₀ + m₁ +.......+ m_k-1 + b₀ = 0 (mod 2). For k = 3, set up the 2^k possible codewords in the code defined by this rule.