One sided noise spectral density VS double sided noise spectral density

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SUMMARY

The discussion clarifies the relationship between one-sided noise spectral density (No) and double-sided noise spectral density (N) in communication systems, establishing that 2N = No. This relationship arises from the differing representations of complex and real signals, where a complex signal like \exp{i\omega_0 t} has a one-sided spectrum, while its real counterpart \cos{\omega_0 t} exhibits a two-sided spectrum. The power spectral density (PSD) of thermal noise is also influenced by whether a real or complex representation is used, with audio engineers typically favoring real representations and communications engineers preferring complex ones.

PREREQUISITES
  • Understanding of one-sided and double-sided noise spectral density
  • Familiarity with power spectral density (PSD) concepts
  • Knowledge of complex and real signal representations
  • Basic principles of communication systems
NEXT STEPS
  • Study the derivation of the relationship between one-sided and double-sided noise spectral densities
  • Learn about power spectral density (PSD) in thermal noise applications
  • Explore the implications of using complex versus real signal representations in communication systems
  • Investigate the role of noise spectral density in audio engineering and communications engineering
USEFUL FOR

This discussion is beneficial for communication engineers, audio engineers, and students studying signal processing who seek to understand the nuances of noise spectral density in different signal representations.

unseensoul
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No is the one sided noise spectral density in communication systems. N is the double sided noise spectral density.

The relationship between them is given by 2N = No but I don't understand why it is like that. Why isn't it N = 2No instead?!
 
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A complex signal like \exp{i\omega_0 t} has a one-sided spectrum, in this case \delta(\omega-\omega_0). Its real counterpart \cos{\omega_0 t} has the two-sided spectrum

\frac{1}{2}[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]

The same power is spread in the real-signal case over positive and negative frequencies, each of which is half as large as the complex spectrum.

When you talk of the power spectral density (PSD) of thermal noise, it again matters whether you are using a real or a complex representation. Audio engineers often use the former, for instance, communications engineers generally the latter. For white noise where the PSD is a constant, and using your notation where N is two-sided and N0 is one-sided, they are related by N0 = 2N.
 

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