Relationship Between E_b/N_0 and \mathtt{SNR}

Relationship Between \(E_b/N_0\) and \(\mathtt{SNR}\)#

\(E_b/N_0\)#

\(E_b\): Energy per Bit#

Definition: Energy is power multiplied by time:

\[ \boxed{ E_b = P_{\text{sig}} \times T_b } \]

Since \( T_b \) (bit duration) is the reciprocal of \( R_b \) (bit rate):

\[ T_b = \frac{1}{R_b} \quad [\text{seconds (s)}] \]

Substituting \( T_b \) into the equation:

\[\boxed{ E_b = \frac{P_{\text{sig}}}{R_b} } \]

where:

  • \( E_b \): Energy per bit [Joules (J) = W/(bits/s)]

  • \( P_{\text{sig}} \): Signal power [Watts (W)]

  • \( T_b \): Bit duration [Seconds (s)]

  • \( R_b \): Bit rate [Bits per second (bps)]

\(N_0\): Noise Power Spectral Density#

Definition: \(N_0\) represents how noise power is distributed over bandwidth:

\[\boxed{ N_0 = \frac{P_{\text{noise}}}{B} } \]

where:

  • \( N_0 \): Noise power spectral density [Watts per Hertz (W/Hz)]

  • \( P_{\text{noise}} \): Total noise power [Watts (W)]

  • \( B \): Bandwidth [Hertz (Hz)]

\(E_b/N_0\): Energy per Bit to Noise Power Spectral Density Ratio#

Combining the definitions of \(E_b\) and \(N_0\):

\[ \frac{E_b}{N_0} = \frac{P_{\text{sig}} \times T_b}{\frac{P_{\text{noise}}}{B}} \quad [\text{dimensionless}] \]

Substituting \(T_b = \frac{1}{R_b}\):

\[ \frac{E_b}{N_0} = \frac{\frac{P_{\text{sig}}}{R_b}}{\frac{P_{\text{noise}}}{B}} \quad [\text{dimensionless}] \]

Thus, we have:

\[\boxed{ \frac{E_b}{N_0} = \frac{P_{\text{sig}}}{P_{\text{noise}}} \times \frac{B}{R_b} } \]

where:

  • \( E_b \): Energy per bit [J]

  • \( N_0 \): Noise power spectral density [W/Hz]

  • \( P_{\text{sig}} \): Signal power [W]

  • \( P_{\text{noise}} \): Noise power [W]

  • \( R_b \): Bit rate [bps]

  • \( B \): Bandwidth [Hz]

  • \(E_b/N_0\): dimensionless

Unit of \(E_b/N_0\)#

The unit of \(E_b/N_0\) (Energy per bit to noise power spectral density ratio) is also dimensionless because:

  • \(E_b\) (Energy per bit) has units of energy \([ \text{Joules (J)} = \text{Watts (W)} \times \text{Seconds (s)} ]\).

  • \(N_0\) (Noise power spectral density) has units of \([ \text{Watts/Hz} = \text{Joules} ]\).

Thus, when dividing \(E_b\) by \(N_0\), the units cancel out:

\[ \frac{E_b}{N_0} = \frac{\text{Joules (J)}}{\text{Joules (J)}} = \text{dimensionless}. \]

Expressing \(E_b/N_0\) in Decibels [dB]

Although \(E_b/N_0\) is dimensionless, it is typically expressed in decibels [dB] for easier analysis in communication systems:

\[ E_b/N_0 \, \text{[dB]} = 10 \log_{10}\left(\frac{E_b}{N_0}\right). \]

Spectral Efficiency (SE)#

Spectral efficiency (SE) [bits/s/Hz] represents the number of bits transmitted per second per unit of bandwidth.

\[\boxed{ \mathtt{SE} = \frac{R_b}{B} } \]

It indicates how efficiently a communication system utilizes its available bandwidth.

Signal-to-Noise Ratio (SNR)#

The Signal-to-Noise Ratio (SNR) quantifies the strength of the signal relative to the background noise:

\[\boxed{ \mathtt{SNR} = \frac{P_{\text{sig}}}{P_{\text{noise}}} } \]

where:

  • \(P_{\text{sig}}\): Signal power \([ \text{Watts (W)} ]\)

  • \(P_{\text{noise}}\): Noise power \([ \text{Watts (W)} ]\)

Unit of SNR#

  • SNR is a dimensionless quantity, as it is a ratio of two powers with the same unit.

  • It is often expressed in decibels [dB] for practical use:

    \[ \mathtt{SNR} \text{[dB]} = 10 \log_{10}\left(\frac{P_{\text{sig}}}{P_{\text{noise}}}\right). \]

Relationship Between \(E_b/N_0\) and SNR#

The relationship between \(E_b/N_0\) (Energy per bit to noise power spectral density ratio) and \(\mathtt{SNR}\) (Signal-to-Noise Ratio) is derived as follows:

\[\boxed{ \frac{E_b}{N_0} = \mathtt{SNR} \times \frac{B}{R_b} } \]

or

\[\boxed{ \mathtt{SNR} = \frac{E_b}{N_0} \times \frac{R_b}{B} } \]

This aligns with the result in [Goldsmith, Eq. (6.1)]

where:

  • \(E_b/N_0\): Energy per bit to noise power spectral density ratio (dimensionless),

  • \(\mathtt{SNR}\): Signal-to-Noise Ratio (dimensionless),

  • \(B\): Bandwidth \([ \text{Hz} ]\),

  • \(R_b\): Bit rate \([ \text{bits/s} ]\).

  • \(\frac{R_b}{B}\): Represents spectral efficiency \([ \text{bits/s/Hz} ]\).

Expressing in dB

SNR in terms of \(E_b/N_0\) is expressed as:

\[ \mathtt{SNR} \, \text{[dB]} = E_b/N_0 \text{[dB]} + 10 \log_{10}\left(\frac{R_b}{B}\right), \]

Key Insights

  1. \(E_b/N_0\) and \(\mathtt{SNR}\) are directly proportional. A higher \(E_b/N_0\) improves the \(\mathtt{SNR}\).

  2. The ratio \(\frac{R_b}{B}\) (spectral efficiency) impacts \(\mathtt{SNR}\). Increasing spectral efficiency decreases \(\mathtt{SNR}\) for the same \(E_b/N_0\).

  3. This relationship highlights the trade-offs between bandwidth, bit rate, and signal quality in a communication system.