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Detection of Periodic Signals in the Presence of Noise (by Autocorrelation)
Detection of Periodic Signals in the Presence of Noise
The noise signal is an unwanted signal which has random amplitude variation. The noise signals are uncorrelated with any periodic signal.
Detection of the periodic signals masked by noise signals is of great importance in signal processing. It is mainly used in the detection of radar and sonar signals, the detection of periodic components in brain signals, in the detection of periodic components in sea wave analysis and in many other areas of geophysics etc. The solution of these problems can be easily provided by the correlation techniques. The autocorrelation function, therefore can be used to detect a periodic signal which masked by a noise signal.
Consider $\mathit{x}\mathrm{(t)}$ is a periodic signal and $\mathit{n}\mathrm{(t)}$ is the noise signal. Then, the correlation function of the signals $\mathit{x}\mathrm{(t)}$ and $\mathit{n}\mathrm{(t)}$ is given by,
$$\mathit{R_\mathit{xn}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\lim_{T \rightarrow \infty}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathit{T/\mathrm{2}}}^{\mathit{T/\mathrm{2}}}\mathit{x}\mathrm{(t)}\mathit{n}\mathrm{(\mathit{t-\mathit{\tau}})}\mathit{dt}\:\mathrm{=}\:\mathrm{0}; \:\:\:\: (for \: all\: \mathit{\tau})$$
Detection of Periodic Signal by Autocorrelation
If a periodic signal $\mathit{x}\mathrm{(t)}$ is mixed with a noise signal $\mathit{n}\mathrm{(t)}$.Then, the received signal is given by,
$$\mathit{y}\mathrm{(t)}\:\mathrm{=}\:\mathit{x}\mathrm{(t)}\mathrm{+}\mathit{n}\mathrm{(t)}$$
The received signal $\mathit{y}\mathrm{(\mathit{t})}$ is also a periodic signal.
Also, let the autocorrelation functions of the signals $\mathit{y}\mathrm{(\mathit{t})}$,$\mathit{x}\mathrm{(\mathit{t})}$ and $\mathit{n}\mathrm{(\mathit{t})}$ are given as follows −
$$\mathit{y}\mathrm{(t)} \:\leftrightarrow\:\mathit{R_{yy}}\mathrm{(\tau)}$$ $$\mathit{x}\mathrm{(t)} \:\leftrightarrow\:\mathit{R_{xx}}\mathrm{(\tau)}$$ $$\mathit{n}\mathrm{(t)} \:\leftrightarrow\:\mathit{R_{nn}}\mathrm{(\tau)}$$
Then, from the definition of autocorrelation function of periodic signals, we have,
$$\mathit{R_\mathit{yy}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\lim_{T \rightarrow \infty}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathit{T/\mathrm{2}}}^{\mathit{T/\mathrm{2}}}\mathit{y}\mathrm{(t)}\mathit{y}\mathrm{(\mathit{t-\mathit{\tau)}}}\mathit{dt}$$ $$\Rightarrow\mathit{R_\mathit{yy}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\lim_{T \rightarrow \infty}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathit{T/\mathrm{2}}}^{\mathit{T/\mathrm{2}}}\mathrm{[\mathit{x}\mathrm{(t)}\mathrm{+}\mathit{n}\mathrm{(t)}]}\mathrm{}[\mathit{x}\mathrm{(\mathit{t-\tau})}]\mathrm{+}\mathit{n}\mathrm{(\mathit{t-\tau})}]\:\mathit{dt}$$ $$\Rightarrow\mathit{R_\mathit{yy}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\lim_{T \rightarrow \infty}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathit{T/\mathrm{2}}}^{\mathit{T/\mathrm{2}}}\mathit{x}\mathrm{(\mathit{t})}\mathit{x}\mathrm{(\mathit{t-\tau})}\mathrm{+}\mathit{n}\mathrm{(\mathit{t})}\mathit{n}\mathrm{(\mathit{t-\tau})}\mathrm{+}\mathit{x}\mathrm{(\mathit{t})}\mathit{n}\mathrm{(\mathit{t-\tau})}\mathrm{+}\mathit{n}\mathrm{(\mathit{t})}\mathit{x}\mathrm{(\mathit{t-\tau})}\mathit{dt}$$ $$\Rightarrow\mathit{R_\mathit{yy}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\lim_{T \rightarrow \infty}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathit{T/\mathrm{2}}}^{\mathit{T/\mathrm{2}}}\mathit{x}\mathrm{(\mathit{t})}\mathit{x}\mathrm{(\mathit{t-\tau})}\mathit{dt}\mathrm{+}\lim_{T \rightarrow \infty}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathit{T/\mathrm{2}}}^{\mathit{T/\mathrm{2}}}\mathit{n}\mathrm{(\mathit{t})}\mathit{n}\mathrm{(\mathit{t-\tau})}\mathit{dt}\mathrm{+}\lim_{T \rightarrow \infty}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathit{T/\mathrm{2}}}^{\mathit{T/\mathrm{2}}}\mathit{x}\mathrm{(\mathit{t})}\mathit{n}\mathrm{(\mathit{t-\tau})}\mathit{dt}\mathrm{+}\lim_{T \rightarrow \infty}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathit{T/\mathrm{2}}}^{\mathit{T/\mathrm{2}}}\mathit{n}\mathrm{(\mathit{t})}\mathit{x}\mathrm{(\mathit{t-\tau})}\mathit{dt}$$ $$\therefore\mathit{R_{yy}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\mathit{R_{xx}}\mathrm{(\tau)}\mathrm{+}\mathit{R_{nn}}\mathrm{(\mathit{\tau})}\mathrm{+}\mathit{R_{xn}}\mathrm{(\mathit{\tau})}\mathrm{+}\mathit{R_{nx}}\mathrm{(\mathit{\tau})}$$
Since, the periodic signal $\mathit{x}\mathrm{(t)}$ and the noise signal $\mathit{n}\mathrm{(t)}$ are uncorrelated. Therefore,
$$\mathit{R_{xn}}\mathrm{(\mathit{\tau})}\:\mathrm{=} \:\mathit{R_{nx}}\mathrm{(\mathit{\tau})}\mathrm{= 0}$$ $$\therefore\mathit{R_{yy}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\mathit{R_{xx}}\mathrm{(\mathit{\tau})}\mathrm{+}\mathit{R_{nn}}\mathrm{(\mathit{\tau})}$$
Hence, the autocorrelation of the mixed signal$\mathit {R_{yy}}\mathrm{(\mathit{\tau})}$ has two components viz.$\mathit{R_{xx}}\mathrm{(\mathit{\tau})}$ and $\mathit{R_{nn}}\mathrm{(\mathit{\tau})}$. As the autocorrelation of a periodic function is also a periodic function of the same frequency while the autocorrelation of an aperiodic function tends to zero for large values of the delay parameter $\mathrm{(\mathit{\tau})}$.
Since in our case the signal $\mathit{x}\mathrm{(\mathit{t})}$ is a periodic signal and the noise signal $\mathit{n}\mathrm{(\mathit{t})}$ is an aperiodic signal. Therefore, the autocorrelation function $\mathit {R_{xx}}\mathrm{(\mathit{\tau})}$ is a periodic function while the autocorrelation function $\mathit {R_{nn}}\mathrm{(\mathit{\tau})}$ becomes very small for large values of the parameter $\tau$. Hence, for large values of the parameter $\mathrm{(\mathit{\tau})}$, we have,
$$\therefore\mathit{R_{yy}}\mathrm{(\mathit{\tau})}\mathrm{=}\mathit{R_{xx}}\mathrm{(\mathit{\tau})}$$
Thus, in this way any periodic signal can be detected in the presence of the noise signal by autocorrelation.
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