Prove that SNR of power = (SNR of voltage) sequare?

To prove that the signal-to-noise ratio (SNR) of power is equal to the square of the SNR of voltage, we can start with the basic definitions of SNR in terms of power and voltage.

SNR in Terms of Power

The SNR in terms of power is defined as the ratio of the average signal power ((P{signal})) to the average noise power ((P{noise})):

[ \text{SNR}{\text{power}} = \frac{P{signal}}{P_{noise}} ]

SNR in Terms of Voltage

When signals are measured in terms of voltage, the SNR is defined as the ratio of the root mean square (RMS) signal voltage ((V{signal})) to the RMS noise voltage ((V{noise})). However, to relate this to power, we must square the voltage ratio because power is proportional to the square of voltage (assuming the same impedance for both signal and noise):

[ \text{SNR}{\text{voltage}} = \frac{V{signal}}{V_{noise}} ]

Since power is proportional to the square of voltage ((P = \frac{V^2}{R}), where (R) is the resistance or impedance), we can express the power SNR in terms of voltage as:

[ \text{SNR}{\text{power}} = \left(\frac{V{signal}}{V_{noise}}\right)^2 ]

This shows that the SNR in terms of power is indeed the square of the SNR in terms of voltage.

Proof

Given:

• (P{signal} = \frac{V{signal}^2}{R}) and (P{noise} = \frac{V{noise}^2}{R}), where (R) is the impedance.

Then: [ \text{SNR}{\text{power}} = \frac{P{signal}}{P{noise}} = \frac{\frac{V{signal}^2}{R}}{\frac{V{noise}^2}{R}} = \left(\frac{V{signal}}{V{noise}}\right)^2 = (\text{SNR}{\text{voltage}})^2 ]

Thus, we have proven that the SNR of power is equal to the square of the SNR of voltage.

Conclusion

The relationship between the SNR of power and the SNR of voltage is fundamentally based on the fact that power is proportional to the square of voltage. This relationship is crucial in understanding how signal strength and noise levels affect communication systems and other applications where signal quality is important.