# Why is the relationship between Es/N0 and SNR different for complex and real signals

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Yaowei Zhu el 11 de Nov. de 2020
Editada: Andrew Reibold el 8 de Mzo. de 2024
I'm confused about some statements in the introduction of AWGN Channel. Here is the link AWGN Channel.
It says that
I wonder why there is a factor of 0.5 for real input signals.
Would you please give me some explanation or introduce me some textbooks that can help me figure out how to derive the equations above?
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Kiran Felix Robert el 18 de Dic. de 2020
Hi Yaowei,
The Symbol rate of a complex input signal is higher when compared to that of a real input signal by a factor of 2.
To understand this, you can think of the complex space as C as being directly mapped to R2 Space. [(x, y) <=> x + i.y]
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Yaowei Zhu el 19 de En. de 2021
I think this is the answer. Thank you so much!
Yaowei Zhu el 4 de Jun. de 2021
Editada: Yaowei Zhu el 12 de Sept. de 2021
In my opinion, no matter it's complex signal or real signal, the SNR equals to the input signal power,S, divided by the noise power, N. The difference is, for complex signal,
S = Si + Sq, (Si = Sq, no consideration of IQ imbalance)
and for real signal,
S = Si or S = Sq
Then, no matter it's complex or real,
Es = S*Tsym
and,
N0 = N*Tsamp
So,
Es/N0 = 10*log10(Tsym/Tsamp) + SNR
Then, the question is why there is a 0.5 factor exists in the Es/N0 formula of real signal as the documentation of MATLAB AWGN CHANNEL model illustrated. I suppose it's because the AWGN CHANNEL just simply think the signal power that is specified through SNR or Eb/N0 or Es/N0 in the AWGN model configuration panel is the power of a complex signal. So, if one want to spcify a SNR of 10dB in case of a real signal, he/she has to set the SNR to 13dB in the configuration panel. Hope I made it clear, English is really not what I'm good at.

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Andrew Reibold el 8 de Mzo. de 2024
Editada: Andrew Reibold el 8 de Mzo. de 2024
Your explanation is quite accurate, and your understanding is correct. The factor of 0.5 in the formula for real input signals is related to how the SNR is specified in MATLAB's AWGN channel model.
In MATLAB, when you specify the SNR for a real signal in the AWGN channel model, it is assumed that the specified SNR corresponds to the power of a complex signal, not a real signal. Since the power of a complex signal is the sum of the powers of its real and imaginary components, for a real signal, you effectively have half the power in each of the real and imaginary parts.
To clarify, let's consider the power relation (Sorry I dont know how to make sub-letters, so I used underscore):
• For a complex signal S=S_i+j*S_q, where S_i and S_q are the real and imaginary parts, the power is ∣S∣^2 = ∣S_i∣^2+∣S_q∣^2∣.
• For a real signal S = S_i, the power is ∣S∣^2=∣S_i|^2 .
Now, if you want the SNR to be specified in terms of the power of the real signal (S_i), you need to adjust the SNR value accordingly. This adjustment involves halving the power, which corresponds to subtracting 3 dB. Hence, the factor of 0.5 in the formula for real input signals.
Your understanding and explanation are correct: if you want to specify an SNR of X dB for a real signal, you would need to set the SNR to X−3 dB in MATLAB's AWGN channel model.
As for textbooks, if you're looking for a comprehensive resource on digital communication systems, "Digital Communications" by John Proakis and Masoud Salehi is a widely used textbook that covers these concepts in detail. It provides a solid foundation in communication theory, including topics like modulation, coding, and noise in communication channels.
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