Signal-to-noise Ratio

Consider an observed trace $R_t$ , consisting of a signal trace $S_t$ and zero-mean independent and identically-distributed noise $n_t$ of variance $\sigma^2$ , as in

$$R_t = S_t + n_t, \quad t=1,\ldots,T.$$

When $M$ such observed traces are drawn and stacked, I get

$$\breve{R_t} \triangleq \sum_{m=1}^{M} R^{(m)}_t$$

$$= \sum_{m=1}^{M}[S_t + n^{(m)}_t]$$

$$= M S_t + \sum_{m=1}^{M} n^{(m)}_t,$$ (71)

where $R^{(m)}_t$ denotes the $m$ th random realization of the signal trace $S_t$ . ($n^{(m)}_t$ 's are still i.i.d.) The signal and the noise part of the stacked trace $\breve{R_t}$ are denoted by

$$\breve{S_t} \triangleq M S_t \quad \textrm{and}$$ (72)

$$\breve{n_t} \triangleq \sum_{m=1}^{M} n^{(m)}_t$$ (73)

respectively. Note that the root mean squared (rms) amplitude of the stacked signal $\breve{S_t}$ is

$$A_M \triangleq \sqrt{\sum_{t=1}^{T}\breve{S_t}^2 /T}$$

$$= M \sqrt{\sum_{t=1}^{T} {S_t}^2 /T}$$

$$= M A_1,$$ (74)

where $A_{1}=\sqrt{\sum_{t=1}^{T}S_{t}^{2}/T}$ is the rms amplitude of the signal trace $S_{t}$ and the second equality follows from equation B.2; and $A_M$ is defined as the rms amplitude of the $M$ -fold stacked signal $\breve{S_t}$ , growing in proportion to $M$ , according to equation B.4. The rms amplitude of the stacked noise $\breve{n_t}$ , $\sigma_M$ , is defined as

$$\sigma_M \triangleq \sqrt{\sum_{t=1}^{T}<\breve{n_t}^2> /T}$$

$$= \sqrt{<\breve{n_t}^2> }$$

$$= \sqrt{ <[\sum_{m=1}^{M} n^{(m)}_t]^2> }$$

$$= \sqrt{ <\sum_{m=1}^{M} {n^{(m)}_t}^2> }$$

$$= \sqrt{M} \sigma,$$ (75)

where $< >$ denotes expectation, the second equality follows because $n_t$ 's are identically-distributed, the third equality follows from equation B.3, the fourth equality follows because $n^{(m)}_t$ 's are zero-mean and independent, and the last equality follows because $n^{(m)}_t$ 's are identically-distributed with variance $\sigma^2$ . Equation B.5 shows that $\sigma_M$ grows in proportion to $\sqrt{M}$ .

Finally, The SNR of $\breve{R_t}$ is defined as the ratio of rms amplitude of signal over that of noise (Papoulis, 1991),

$$\textrm{SNR} \triangleq \frac{ A_M } { \sigma_M }$$

$$= \frac{ M A_1 } { \sqrt{M} \sigma }$$

$$= \sqrt{M} A_1 / \sigma,$$ (76)

which exhibits a $\sqrt{M}$ enhancement.