I have found two kinds of formulas to calculate the capacity of the AWGN channel with binary input. The first well-known formula is log (1+SNR) bits per channel use, the second one is half the first one, i.e. 1/2 log (1+SNR) bits per channel use. I have one interpretation, which the first formula is for the complex-valued AWGN channel, and the second formula is for the real-valued AWGN channel.
Can you confirm my interpretation or have you another interpretation?
See this
https://web.stanford.edu/~dntse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter5.pdf
This is an interesting question. It sounds reasonable that when a band-limited channel is used to transmit a real signal, the channel capacity (CC) is half compared to transmitting a complex signal. However, we should be careful to remember that the Fourier transform (FT) of a real (baseband) signal is symmetrical around DC (i.e. f=0), but the FT of a complex (baseband) signal is unsymmetrical in general. Thus, when an RF channel is used, the spectrum of a real (baseband) signal is symmetrical about the RF carrier frequency. As a result, it is possible to transmit only half of the signal spectrum, by using the so-called single-side band (SSB) transmission. On the other hand, in order to transmit a complex signal, full spectrum has to be transmitted since its spectrum is unsymmetrical. Thus, when a SSB transmitted is used to transmit a real signal, the channel capacity is the same as to transmit a complex signal.
Summary:
(1) When the double-side band (DSB) transmission is used for transmission of both real and complex signals, the channel capacity is half for the real signal.
(2) However, when SSB is used for transmission of real signals while DSB has to be used for transmission of complex signals, the channel capacity is the same for both real signals and complex signals.
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