... using complexReX[k] ImX[k]substitution, the frequency domain can be represented by a single complexarray, . In the complex Fourier transform, both & are arraysX[k] x[n] X[k]of complex numbers. ... aperiodic. 2. Real versus ComplexEach of these four transforms has a complex version and a real version. Thecomplex versions have a complex time domain signal and a complex frequencydomain signal. ... 'ejx% e&jx2sin(Tt) '12jej(&T)t&12jejTtEQUATION 31-4Sinusoids as complex numbers. Usingcomplex numbers, cosine and sine wavescan be written as the sum of a positiveand a negative...