The IQ signal x_c(t) has associated complex envelope,

Note that you can get the original signal x_c(t) by multiplying (spinning) the complex envelope by the complex sinusoid

Transmitter xt N pin DAC X. xin) x) rid) cos, Source Data Bits Radio Channel pin DAC Lower path not used for BPSK wit) WGN Insert, zeros values Co- & adjacent Enxol + ixon channel signals -I-N/T-NR. via radio channel it N-samples per symbol Pulse shape r(t) = x +xir) wit) ADC 71 n] BPF 71 77 Matched Filter Symbol Decisions 2 cpx 2 LPF 2 cos2x18+) 2 sin2x4+) LPF Frequency & phase errors present without synchronization ADC del pl2MN- flipped & shifted tx pulse Receiver X(t) To get this signal, rely on the extended version of the phasor addition formula: x.(t)= x,(t)cos (21f.t) xo(t)sin(21f_t) = x,(t)cos(2nf_t)+(-xa(t))cos (21fct-1/2) =R(t)cos(21f.t + (t)) Here, the complex envelope, R(t)ejo(t) = (t) is given by R(t)eltt = (t)= x(t)+xx(t):e" els/= x(t)+ jxx(t) X(t) To get this signal, rely on the extended version of the phasor addition formula: x.(t)= x,(t)cos (21f.t) xo(t)sin(21f_t) = x,(t)cos(2nf_t)+(-xa(t))cos (21fct-1/2) =R(t)cos(21f.t + (t)) Here, the complex envelope, R(t)ejo(t) = (t) is given by R(t)eltt = (t)= x(t)+xx(t):e" els/= x(t)+ jxx(t) Transmitter xt N pin DAC X. xin) x) rid) cos, Source Data Bits Radio Channel pin DAC Lower path not used for BPSK wit) WGN Insert, zeros values Co- & adjacent Enxol + ixon channel signals -I-N/T-NR. via radio channel it N-samples per symbol Pulse shape r(t) = x +xir) wit) ADC 71 n] BPF 71 77 Matched Filter Symbol Decisions 2 cpx 2 LPF 2 cos2x18+) 2 sin2x4+) LPF Frequency & phase errors present without synchronization ADC del pl2MN- flipped & shifted tx pulse Receiver X(t) To get this signal, rely on the extended version of the phasor addition formula: x.(t)= x,(t)cos (21f.t) xo(t)sin(21f_t) = x,(t)cos(2nf_t)+(-xa(t))cos (21fct-1/2) =R(t)cos(21f.t + (t)) Here, the complex envelope, R(t)ejo(t) = (t) is given by R(t)eltt = (t)= x(t)+xx(t):e" els/= x(t)+ jxx(t) X(t) To get this signal, rely on the extended version of the phasor addition formula: x.(t)= x,(t)cos (21f.t) xo(t)sin(21f_t) = x,(t)cos(2nf_t)+(-xa(t))cos (21fct-1/2) =R(t)cos(21f.t + (t)) Here, the complex envelope, R(t)ejo(t) = (t) is given by R(t)eltt = (t)= x(t)+xx(t):e" els/= x(t)+ jxx(t)