Signal Power Calculation

To determine the Signal-to-Noise Ratio (SNR), our first step involves calculating the signal power that reaches the receiver after passing through the transmission medium.

The initial signal power ($P_{in}$) is provided as:

$P_{in} = 1 \text{ mW}$

We must convert this power unit from milliwatts (mW) to Watts (W) for consistency in calculations:

$P_{in} = 1 \times 10^{-3} \text{ W}$

The signal experiences a loss of 40 dB through the cable. A decibel (dB) represents a logarithmic ratio. A loss of 40 dB indicates that the signal power is significantly reduced. The factor by which the power is reduced can be calculated using the formula $10^{(Loss_{dB}/10)}$:

The power reduction factor due to the cable loss is calculated as:

$$ \text{Loss Factor} = 10^{(40 \text{ dB} / 10)} = 10^4 $$

Now, we find the signal power at the receiver ($P_{rx}$) by dividing the initial signal power by this loss factor:

$$ P_{rx} = \frac{P_{in}}{\text{Loss Factor}} $$

$$ P_{rx} = \frac{1 \times 10^{-3} \text{ W}}{10^4} $$

$$ P_{rx} = 1 \times 10^{-7} \text{ W} $$

Noise Power Calculation

The next crucial step is to calculate the total noise power present in the signal's bandwidth at the receiver.

We are given the effective one-sided noise spectral density ($N_0$), which represents the noise power per unit of bandwidth:

$N_0 = 10^{-20} \text{ Watt/Hz}$

The signal has a bandwidth ($B$) of 100 MHz. We convert this bandwidth to Hertz (Hz) for our calculations:

$B = 100 \text{ MHz} = 100 \times 10^6 \text{ Hz} = 1 \times 10^8 \text{ Hz}$

The total noise power ($N$) within this bandwidth is found by multiplying the noise spectral density ($N_0$) by the bandwidth ($B$):

$$ N = N_0 \times B $$

$$ N = (10^{-20} \text{ Watt/Hz}) \times (1 \times 10^8 \text{ Hz}) $$

$$ N = 1 \times 10^{-12} \text{ W} $$

SNR Calculation Steps

With both the received signal power and the total noise power determined, we can now proceed to calculate the Signal-to-Noise Ratio (SNR).

Step 1: Signal Power Conversion

The initial signal power was converted from $1 \text{ mW}$ to $1 \times 10^{-3} \text{ W}$. This ensures all power values are in the same base unit (Watts).

Step 2: Received Signal Power

After applying the 40 dB cable loss, the signal power reaching the receiver was calculated:

$P_{rx} = 1 \times 10^{-7} \text{ W}$

Step 3: Noise Power Calculation

The total noise power within the specified $100 \text{ MHz}$ bandwidth was calculated using the noise spectral density:

$N = 1 \times 10^{-12} \text{ W}$

Step 4: Linear SNR Calculation

The Signal-to-Noise Ratio (SNR) in its linear form is obtained by dividing the received signal power ($P_{rx}$) by the noise power ($N$):

$$ SNR_{linear} = \frac{P_{rx}}{N} $$

$$ SNR_{linear} = \frac{1 \times 10^{-7} \text{ W}}{1 \times 10^{-12} \text{ W}} $$

$$ SNR_{linear} = 1 \times 10^{5} $$

Step 5: SNR to Decibels Conversion

Finally, to express the SNR in decibels (dB), which is common practice, we use the following logarithmic formula:

$$ SNR_{dB} = 10 \times \log_{10}(SNR_{linear}) $$

Substituting the calculated linear SNR:

$$ SNR_{dB} = 10 \times \log_{10}(1 \times 10^{5}) $$

The base-10 logarithm of $1 \times 10^{5}$ is simply 5.

$$ SNR_{dB} = 10 \times 5 $$

$$ SNR_{dB} = 50 \text{ dB} $$

Final Signal-to-Noise Ratio

The Signal-to-Noise Ratio (SNR) at the receiver, after considering the initial signal power, the cable loss, the signal bandwidth, and the noise spectral density, is determined to be 50 dB.