One of the main challenges in beamforming is how to isolate unwanted signals and attenuate them as much as possible. In some cases, the noise is correlated with the signal of interest, making it the more challenging. However, in most cases, the noise at the microphones being utilized in beamforming are independent and identically distributes whilst being uncorrelated with the signal of interest. In such cases, most applications rely on the low computational algorithm of spectral estimation and subtraction. We present a brief description of noise suppression using spectral subtraction.

Consider an acoustic signal impinging a microphones. suppose the signal at the microphone, $x$ , can be denoted as $x(t) = s(t) + \nu(t)$

where both the nonlinear attenuation and delay have been subsumed without any loss of generality in $s(t)$,the source signal, and $\nu(t)$ is the noise. Both $s(t)$ and $\nu (t)$ are zero mean ergodic processes. The frequency domain representation becomes $X(w) = S(w) +N(w), N(w) \sim \mathbb{N} (0,\sigma_n^2)$

The expectation of the squared magnitude response can be expressed as: $|X(w)|^2 = |S(w)|^2 +|N(w)|^2 +2\mathbb{E}\left[S(w) N^{*}(w)\right]$

Since the noise term and the signal of interest are uncorrelated, the expectation of the cross-term diminishes to zero leaving $|S(w)|^2 \approx |X(w)|^2 -|N(w)|^2$

Now, suppose there exists a noisy estimate of the noise spectrum, for example during periods of only noise, then the received signal of interest will approximate $|\hat{S}(w)|^2 \approx |X(w)|^2\left( 1 -\frac{|\hat{N}(w)|^2}{|X(w)|^2} \right)$

Further, suppose a noisy estimate of the phase of the signal of interest, $\hat{\theta}$, is known, then, the received signal is estimated by using $|\hat{S}(w)| e^{jw\hat{\theta}} \approx \hat{S}(w) = e^{jw\hat{\theta}} |X(w)| \sqrt{\left( 1 -\frac{|\hat{N}(w)|^2}{|X(w)|^2} \right)}$ $\hat{S}(w) \approx X(w) \sqrt{\left( 1 -\frac{|\hat{N}(w)|^2}{|X(w)|^2} \right)}$

Note that the square root term has to be real and is set to zero when the value becomes a complex number.

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