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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.
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1  
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 –  endolith Jun 5 '12 at 14:35
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4 Answers

up vote 8 down vote accepted

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).

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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. –  endolith Jun 5 '12 at 14:40
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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. –  aagaard Jun 5 '12 at 15:04
    
Well for me, it never made any sense where the arbitrary resistor values came from. –  endolith Jun 5 '12 at 15:24
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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.

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A direct answer to your first question:

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 $x[n]$.

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Even though the 20*log(ratio of arbitrary quantities) form is almost ubiquitously used when defining SNR in the DSP realm, I maintain it is incorrect, and the 10*log(ratio) should always be used. The definition of dB is 10*log(ratio of powers). In the analog realm, power is V^2/R, where V and R are voltage and impedence across that voltage respectively, so we have dB = 10*log((V1^2/R1) / (V2^2/R2)) = 20*log(V1/V2) + 10*log(R2/R1). Especially at high frequencies (radar, communications, etc), it is generally desirable to match impedences for maximum transfer of power, meaning that R2 = R1. If that is the case then the second term vanishes and we're left with this definition of dB: 20*log(V1/V2). Perfectly acceptable when dealing with analog circuits. But when working with software in the DSP realm, what is an impedence? What is a voltage? Those concepts have no meaning, they do not exist in software, they are HARDWARE concepts. Once we have converted an analog signal to digital, the concepts of voltage and impedence are meaningless. So we can hardly say that impedences match (what would that even mean? the impedence of the circuit sourcing the ADC matches the input impedence of the ADC? LUDICROUS!!), meaning we have no basis for saying that the second term vanishes, and thus no basis for defining a dB as 20*log(ratio of arbitrary quantities). Therefore, it makes most sense to revert to the definition of a dB and use the 10*log(ratio of arbitrary quantities) form.

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Your answer is a little waffly - it would be preferable if you could answer the question more directly (and please break up your paragraphs!) –  PAK-9 Nov 20 '13 at 19:21
    
OK PAK, how about this (although it should have been obvious from my first answer): the authors PROBABLY mean 20dB = 20log(Es/Ew), meaning that Es has 10 times more power than Ew. HOWEVER, if the definition of SNR had been used correctly (i.e. the authors likely used it incorrectly), then the meaning would have been 20dB = 10log(Es/Ew), implying Es has 100 times more power than Ew. And although this question is the tip of a very large iceberg, I'll refrain from writing anymore, I don't want to stress you further with my "non-broken-up" paragraphs. –  T B Nov 20 '13 at 21:18
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