# What is meant by 20 dB signal-to-noise ratio?

I'm reading a paper in which a discrete signal

$$x(n) = s(n) + w(n)$$

is considered. $s(n)$ is a known, deterministic series, and $w(n)$ is white noise with zero mean. The authors write that

signals were generated with an SNR of 20 dB

What does this mean?

• What does signal energy mean? There seems to be several ways of defining this, but no attempt to do so is done in this paper.
• What does 20 dB SNR mean? I suppose that either $20=10\log{E_s/E_w}$, or $20=20\log{E_s/E_w}$, but am not familiar enough with these concepts to be sure.
• The 10⋅log10 definition is used for power ratios. The 20⋅log10 definitions is used for amplitudes or "field quantities" like voltage or pressure, since you have to square them first to produce a power ratio. See en.wikipedia.org/wiki/Decibel#Definition Jun 5, 2012 at 14:35
• if SNR=47db then you want to convert into normal value then take antilog(47/10) May 30, 2018 at 17:25

A discrete signal is often interpreted as an analog voltage signal. The signal x(n) should be considered as a sampled version of x(t) with a unknown sampling frequency.

Q1:

So when we're talking about the energy of the signal, it is presumed that the voltage signal is obtained in a circuit with an 1Ω resistor in series, so that v(t)^2/R = P.

Q2:

Signal-to-noise ratio is used in many areas and is defined as the logarithmic power ratio of the signal and noise, i.e. SNR = 10*log(Ps/Pw). The other SNR = 20*log(Ps/Pw) is not true, because the Ps and Pw are power values. The latter is used when the signals are represented as "voltage" signals, i.e. SNR = 20*log(s(n)/w(n)) = 10*log(s(n)^2/w(n)^2).

• There's no assumption of 1 ohm or even voltage. It could be 100 ohm or any other electrical resistance (for a voltage signal) or any acoustic impedance (for a sound pressure signal) or anything like that, and the value will cancel out when you calculate the power ratio. Jun 5, 2012 at 14:40
• You're right. Nevertheless, I think it is easier to understand and interpret with that assumption. If your were to calculate the energy for $s(n)$ and $w(n)$ separately with a given sampling frequency it is easier to scale, when you know your actually setup. Jun 5, 2012 at 15:04
• Well for me, it never made any sense where the arbitrary resistor values came from. Jun 5, 2012 at 15:24

SNR is the ratio of the (mean) power of two independent signals, one called "signal" and the other "noise".

If the deterministic signal is periodic, then its power is defined as energy per period $E_s / T_s$.

SNR is normally expressed in dB: $SNR_{dB} = 10 \log P_s/P_w$.

In your particular example, 20 dB means that the signal has 100 times the power of the (interfering) noise.

Energy in a sequence $x[n] = \sum\lvert x[n]\rvert^2$. Which is, according to Parseval's theorem, the same as $\frac{\sum\lvert X(f)\rvert^2}{N}$, where $X(f)$ is the discrete Fourier transform of the length-$N$ sequence $x[n]$.