# How to calculate The Signal-to-Noise Ratio (SNR) in dB (different units)?

Recently, I've ventured into a new domain that I intend to incorporate into my work. After thorough research and consultation, I've found this model to calculate the data transmission rate (Capacity).

The data transmission rate for offloading task $$i \in I$$ to server $$v \in V$$ (volunteer or cloud) is denoted as $$r_{iv}$$. We calculate it as follows:

• $$B_i$$: Wireless bandwidth of the corresponding end-device (Mbit/s).
• $$P_i$$: Transmission power in dBm.
• $$d_{ij}$$: Distance between the user device and server (either volunteer or cloud) in km or meters.

Noise Power

First, compute the noise power, $$\omega_0$$, in dBm: $$\omega_0 = -174 + 10 \log_{10}(B_i)$$

Channel Gain

Next, the channel gain, $$h$$, in dB is computed as: $$h = 10 \log_{10} \left( \frac{\lambda^2}{(4\pi)^2} \right) - 20 \log_{10}(d_{ij})$$ Where $$\lambda$$ is the wavelength (e.g., 0.1 meters).

SNR in dB

The Signal-to-Noise Ratio (SNR) in dB is: $$\text{SNR}_{\text{dB}} = P_i + h - \omega_0$$

Conversion of SNR from dB to Linear Scale

$$\text{SNR}_{\text{linear}} = 10^{\frac{\text{SNR}_{\text{dB}}}{10}}$$

Transmission Rate

Finally, the transmission rate, $$r_{iv}$$, is: $$r_{iv} = B_i \log_2(1 + \text{SNR}_{\text{linear}})$$

**Now, I believe there's an issue: In the formula for $$\text{SNR}_{\text{dB}}$$, we are adding $$\omega_0$$(measured in dBm) and $$h$$ (measured in dB). Can we directly sum these quantities due to their differing units? I would greatly appreciate any insights or feedback from you all. I've invested a significant amount of time in this endeavor and have done my utmost to ensure its accuracy and relevance.

It is like If we have 1 kg of apples and 1g of apples, we simply cannot sum up because 1 kg + 1 g = 2 ??? makes no sense.

You can sum up either kg or g: 1 kg + 0.001 kg = 1.001 kg 1000 g + 1 g = 1001 g

If $$SNR_{db}$$ is in dB, then $$\omega$$, which is in dBm has the wrong unit.

**

## 1 Answer

Yes, you can just add these quantities. $$\omega_0$$ is just a gain. In linear domain, this would be a factor, in logarithmic domain, this transforms to a summand.

• Sorry, I think the answer does not address the issue. If you have 1 kg of apples and 1g of apples, you simply cannot sum up because 1 kg + 1 g = 2 ??? makes no sense. You can sum up either kg or g: 1 kg + 0.001 kg = 1.001 kg 1000 g + 1 g = 1001 g If SNR_db is in dB, then omega, which is in dBm has the wrong unit. Oct 20, 2023 at 19:26
• You need to take a look at the definition of "dB". It is NOT a physical unit. It is always a RATIO. "dBm" means "regarding 1 mW", "dB" means "regarding 1". Both are just factors. And they are expressed logarithmically, so, yes, you can just sum them up, which corresponds to a multiplication in linear domain.
– Max
Oct 21, 2023 at 14:01
• Great dear. Thanks Oct 22, 2023 at 14:59