Symbol interval T=1/(2B); symbol rate = 1/T=2B transmissions/sec (The transmitted baseband signal is assumed to be real here) Noise power = (N_0/2)(2B)=N_0B

\Gamma is no smaller than 1 The encoded PAM or QAM here means that channel coding is applied (e.g., block coding or convolutional coding) such that the attainable data rate can be increased for fixed BER, i.e., \Gamma can be lowered.

symbol interval = T, i.e., symbol rate =1/T =(\Delta f); \phi(t) is the baseband pulse shaping function with bandwidth (\Delta f)/2 With this choice of f_n, the nth subchannel has the frequency band =[(n-1/2)(\delta f),(n+1/2)(\delta f)]. =(\Delta f)/2 H(f)=H*(-f) (Hermitian) Only N symbols are transmitted over N respective orthogonal subchannels in synchronization manner (without ISI). A_n=S_I+jS_Q such as M-ary QAM and M-ary PSK

approximation Y_n=(S_I+jS_Q)+W_n Approximately equals (by (A)) AWGN with psd N_0/2

assuming that H_n is known in advance (e.g., estimated by training signal) Symbol S_i is detected if Y_n\in Z_i for subchannel n channel approximation error rather than ISI channel approximation error rather than ISI The multichannel coherent ML detector above is for the scenario that only one symbol in each subchannel is transmitted in the synchronous manner (without the ISI concern).

{\sum_n R_n (2B/N)} bits/sec (where 2B/N is transmissions/sec for each n) =[{\sum_n R_n} (2B/N)]/2B bits/transmission The channel here is a real lowpass channel of bandwidth B: \Delta f =B/N; \sigma_n^2=(n_0/2x\delta f)x2; (due to positive frequency and negative frequency)

bottom level of subchannel n water level =P Convex optimization (to be introduced in COM5245, "Optimization for Communications") and has been recognized as an important tool for solving many communications and signal processing problems and many other science and engineering problems.

P_n=0 for n=2 and 7 water level bottom level of subchannel n h(t) is causal complex 1/T_s=2B, B is the bandwidth of the complex lowpass signal s(t) to be transmitted

Complex AWGN with psd N_0/2 both for the real part and imaginary part. i.i.d. zero-mean Gaussian with variance 2 N_0B (due to analog lowpass filtering with passband B) in ADC (analog-to-digital conversion) approximately equals

One extended symbol interval The interval [-\nu, N-1+\nu] on which the received signal resides. linear convolution = circular convolution for n=0, 1, 2,..., N-1. m=n-\nu k=1,2,..., N-1

i.i.d. complex Gaussian with zero mean and variance N(2N_0B). Since s(t) is complex, it can be expressed as s(t) = s_i(t) + j s_q(t) (with bandwidth B) that need be up-converted to the real bandpass signal (with bandwidth 2B) s_i(t) cos(2\pi f_ct)-s_q(t) sin(2\pi f_ct) for transmission. The guard interval must be discarded before N-point DFT is performed

Up-conversion: Real bandpass signal for transmission s_i(t) cos(2\pi f_ct)-s_q(t) sin(2\pi f_c t) where f_c>b=1/(2t_s). Its bandwidth is 2B=1/T_s, and the symbol interval T=NT_s (not including the guard interval) converts the received real bandpass signal into the associated baseband signal. Standards using OFDM: WLAN (hundreds of meters): IEEE 802.11a, IEEE 802.11g, IEEE 802.11n (WiFi); Short range and indoors (high bandwidth): MB-OFDM-UWB; Metropolitan: IEEE 802.16 (WiMAX), LTE (3GPP), LTE-advanced (LTE+); DAB, DAB+, DVB-T, DVB-H

unknown variables frequency flat & time flat channel (one path channel h(\tau)=exp{j\theta})

unknown variables x_{1k}(\tau)+j x_{2k}(\tau) gradient e[n] The maximum occurs when (\tilde x_k) (\hat a_k^*) exp{-j\theta}>0

a constant for all k Only pay attention to this part for this page.

numerical approximation

unknown parameter (in [0,T]) to be estimated

\tau=\beta =\hat \tau E[ \tilde x(nt+\tau) ^2] = [(a^2(\tau) +b^2(\tau)] E +N_0 = (1-2a(\tau)+2a^2(\tau)) E + N_0 which is maximized for a(\tau)=1 or the optimum \hat \tau=\beta. Therefore, \hat \tau = arg max E[ \tilde x(nt+\tau) ^2] = \beta ==> Problem (B) which is eaxctly (6.280). \hat \tau = \beta