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Fixing typo that replace omega by t
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lennon310
lennon310
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Let there be two microphones, $x_1(t), x_2(t)$ capturing a source, $s(t)$, with some additive noise $n_{1,2}(t)$ that is uncorrelated with the source and with each other (i.e., its effect can be ignored while calculating the cross-correlation). Let $\tau_0$ (in seconds) be the time difference of arrival between the two microphones. The time domain equations of the microphones are $$x_1(t) = s(t) + n_1(t) \\ x_2(t) = \alpha_0 s(t - \tau_0) + n_2(t) $$

Now, taking the FFT and converting these to the frequency domain would give us, $$ X_1(\omega) = S(\omega) + N_1(\omega) \\ X_2(\omega) = \alpha_0 S(\omega) \exp{(-j\omega \tau_0)} + N_2(\omega) $$

The cross correlation function of the two microphone signals (or the cross-power spectrum in the frequency domain), can be written as

$$ \begin{aligned} R_{x_1,x_2}(l) &= \mathbb{E}[x_1(t) x_2(t-l)] \\ R_{x_1, x_2}(\omega) &= X_1(\omega)X_2^*(\omega) \\ &= |S(\omega)|^2 \alpha_0 \exp{(j\omega \tau_0)} \end{aligned} $$

The Generalized Cross-Correlation function with Phase Transform (GCC-PHAT) at lag $l$ is defined as $$ \begin{aligned} \tilde{R}_{x_1,x_2}(l) =& \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{R_{x_1,x_2}(\omega)}{|R_{x_1,x_2}(\omega)|}e^{j\omega l}d\omega \\ =& \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{j\omega(l + \tau_0)} d\omega \\ &= \delta(l - \tau_0) \end{aligned} $$ where $\delta$ is the Dirac-delta function which is non-zero at $l = \tau_0$, which is the time-difference of arrival (TDOA). Typically, the lag is calculated in samples, so to convert that to seconds, you have to divide it by the sampling rate.

I hope this helps in understanding GCC-PHAT, which is useful tool for TDOA estimation, and hence, source localization.

Let there be two microphones, $x_1(t), x_2(t)$ capturing a source, $s(t)$, with some additive noise $n_{1,2}(t)$ that is uncorrelated with the source and with each other (i.e., its effect can be ignored while calculating the cross-correlation). Let $\tau_0$ (in seconds) be the time difference of arrival between the two microphones. The time domain equations of the microphones are $$x_1(t) = s(t) + n_1(t) \\ x_2(t) = \alpha_0 s(t - \tau_0) + n_2(t) $$

Now, taking the FFT and converting these to the frequency domain would give us, $$ X_1(\omega) = S(\omega) + N_1(\omega) \\ X_2(\omega) = \alpha_0 S(\omega) \exp{(-j\omega \tau_0)} + N_2(\omega) $$

The cross correlation function of the two microphone signals (or the cross-power spectrum in the frequency domain), can be written as

$$ \begin{aligned} R_{x_1,x_2}(l) &= \mathbb{E}[x_1(t) x_2(t-l)] \\ R_{x_1, x_2}(\omega) &= X_1(\omega)X_2^*(\omega) \\ &= |S(\omega)|^2 \alpha_0 \exp{(j\omega \tau_0)} \end{aligned} $$

The Generalized Cross-Correlation function with Phase Transform (GCC-PHAT) at lag $l$ is defined as $$ \begin{aligned} \tilde{R}_{x_1,x_2}(l) =& \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{R_{x_1,x_2}(\omega)}{|R_{x_1,x_2}(\omega)|}e^{j\omega l}d\omega \\ =& \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{j\omega(l + \tau_0)} d\omega \\ &= \delta(l - \tau_0) \end{aligned} $$ where $\delta$ is the Dirac-delta function which is non-zero at $l = \tau_0$, which is the time-difference of arrival (TDOA). Typically, the lag is calculated in samples, so to convert that to seconds, you have to divide it by the sampling rate.

I hope this helps in understanding GCC-PHAT, which is useful tool for TDOA estimation, and hence, source localization.

Let there be two microphones, $x_1(t), x_2(t)$ capturing a source, $s(t)$, with some additive noise $n_{1,2}(t)$ that is uncorrelated with the source and with each other (i.e., its effect can be ignored while calculating the cross-correlation). Let $\tau_0$ (in seconds) be the time difference of arrival between the two microphones. The time domain equations of the microphones are $$x_1(t) = s(t) + n_1(t) \\ x_2(t) = \alpha_0 s(t - \tau_0) + n_2(t) $$

Now, taking the FFT and converting these to the frequency domain would give us $$ X_1(\omega) = S(\omega) + N_1(\omega) \\ X_2(\omega) = \alpha_0 S(\omega) \exp{(-j\omega \tau_0)} + N_2(\omega) $$

The cross correlation function of the two microphone signals (or the cross-power spectrum in the frequency domain), can be written as

$$ \begin{aligned} R_{x_1,x_2}(l) &= \mathbb{E}[x_1(t) x_2(t-l)] \\ R_{x_1, x_2}(\omega) &= X_1(\omega)X_2^*(\omega) \\ &= |S(\omega)|^2 \alpha_0 \exp{(j\omega \tau_0)} \end{aligned} $$

The Generalized Cross-Correlation function with Phase Transform (GCC-PHAT) at lag $l$ is defined as $$ \begin{aligned} \tilde{R}_{x_1,x_2}(l) =& \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{R_{x_1,x_2}(\omega)}{|R_{x_1,x_2}(\omega)|}e^{j\omega l}d\omega \\ =& \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{j\omega(l + \tau_0)} d\omega \\ &= \delta(l - \tau_0) \end{aligned} $$ where $\delta$ is the Dirac-delta function which is non-zero at $l = \tau_0$, which is the time-difference of arrival (TDOA). Typically, the lag is calculated in samples, so to convert that to seconds, you have to divide it by the sampling rate.

I hope this helps in understanding GCC-PHAT, which is useful tool for TDOA estimation, and hence, source localization.

Fixing typo that replace omega by t
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edit approved May 25, 2023 at 15:14
Asimonu
edit approved May 25, 2023 at 15:14
Asimonu
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Let there be two microphones, $x_1(t), x_2(t)$ capturing a source, $s(t)$, with some additive noise $n_{1,2}(t)$ that is uncorrelated with the source and with each other (i.e., its effect can be ignored while calculating the cross-correlation). Let $\tau_0$ (in seconds) be the time difference of arrival between the two microphones. The time domain equations of the microphones are $$x_1(t) = s(t) + n_1(t) \\ x_2(t) = \alpha_0 s(t - \tau_0) + n_2(t) $$

Now, taking the FFT and converting these to the frequency domain would give us, $$ X_1(\omega) = S(\omega) + N_1(\omega) \\ X_2(\omega) = \alpha_0 S(\omega) \exp{(-j\omega \tau_0)} + N_2(t) $$$$ X_1(\omega) = S(\omega) + N_1(\omega) \\ X_2(\omega) = \alpha_0 S(\omega) \exp{(-j\omega \tau_0)} + N_2(\omega) $$

The cross correlation function of the two microphone signals (or the cross-power spectrum in the frequency domain), can be written as

$$ \begin{aligned} R_{x_1,x_2}(l) &= \mathbb{E}[x_1(t) x_2(t-l)] \\ R_{x_1, x_2}(\omega) &= X_1(\omega)X_2^*(\omega) \\ &= |S(\omega)|^2 \alpha_0 \exp{(j\omega \tau_0)} \end{aligned} $$

The Generalized Cross-Correlation function with Phase Transform (GCC-PHAT) at lag $l$ is defined as $$ \begin{aligned} \tilde{R}_{x_1,x_2}(l) =& \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{R_{x_1,x_2}(\omega)}{|R_{x_1,x_2}(\omega)|}e^{j\omega l}d\omega \\ =& \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{j\omega(l + \tau_0)} d\omega \\ &= \delta(l - \tau_0) \end{aligned} $$ where $\delta$ is the Dirac-delta function which is non-zero at $l = \tau_0$, which is the time-difference of arrival (TDOA). Typically, the lag is calculated in samples, so to convert that to seconds, you have to divide it by the sampling rate.

I hope this helps in understanding GCC-PHAT, which is useful tool for TDOA estimation, and hence, source localization.

Let there be two microphones, $x_1(t), x_2(t)$ capturing a source, $s(t)$, with some additive noise $n_{1,2}(t)$ that is uncorrelated with the source and with each other (i.e., its effect can be ignored while calculating the cross-correlation). Let $\tau_0$ (in seconds) be the time difference of arrival between the two microphones. The time domain equations of the microphones are $$x_1(t) = s(t) + n_1(t) \\ x_2(t) = \alpha_0 s(t - \tau_0) + n_2(t) $$

Now, taking the FFT and converting these to the frequency domain would give us, $$ X_1(\omega) = S(\omega) + N_1(\omega) \\ X_2(\omega) = \alpha_0 S(\omega) \exp{(-j\omega \tau_0)} + N_2(t) $$

The cross correlation function of the two microphone signals (or the cross-power spectrum in the frequency domain), can be written as

$$ \begin{aligned} R_{x_1,x_2}(l) &= \mathbb{E}[x_1(t) x_2(t-l)] \\ R_{x_1, x_2}(\omega) &= X_1(\omega)X_2^*(\omega) \\ &= |S(\omega)|^2 \alpha_0 \exp{(j\omega \tau_0)} \end{aligned} $$

The Generalized Cross-Correlation function with Phase Transform (GCC-PHAT) at lag $l$ is defined as $$ \begin{aligned} \tilde{R}_{x_1,x_2}(l) =& \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{R_{x_1,x_2}(\omega)}{|R_{x_1,x_2}(\omega)|}e^{j\omega l}d\omega \\ =& \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{j\omega(l + \tau_0)} d\omega \\ &= \delta(l - \tau_0) \end{aligned} $$ where $\delta$ is the Dirac-delta function which is non-zero at $l = \tau_0$, which is the time-difference of arrival (TDOA). Typically, the lag is calculated in samples, so to convert that to seconds, you have to divide it by the sampling rate.

I hope this helps in understanding GCC-PHAT, which is useful tool for TDOA estimation, and hence, source localization.

Let there be two microphones, $x_1(t), x_2(t)$ capturing a source, $s(t)$, with some additive noise $n_{1,2}(t)$ that is uncorrelated with the source and with each other (i.e., its effect can be ignored while calculating the cross-correlation). Let $\tau_0$ (in seconds) be the time difference of arrival between the two microphones. The time domain equations of the microphones are $$x_1(t) = s(t) + n_1(t) \\ x_2(t) = \alpha_0 s(t - \tau_0) + n_2(t) $$

Now, taking the FFT and converting these to the frequency domain would give us, $$ X_1(\omega) = S(\omega) + N_1(\omega) \\ X_2(\omega) = \alpha_0 S(\omega) \exp{(-j\omega \tau_0)} + N_2(\omega) $$

The cross correlation function of the two microphone signals (or the cross-power spectrum in the frequency domain), can be written as

$$ \begin{aligned} R_{x_1,x_2}(l) &= \mathbb{E}[x_1(t) x_2(t-l)] \\ R_{x_1, x_2}(\omega) &= X_1(\omega)X_2^*(\omega) \\ &= |S(\omega)|^2 \alpha_0 \exp{(j\omega \tau_0)} \end{aligned} $$

The Generalized Cross-Correlation function with Phase Transform (GCC-PHAT) at lag $l$ is defined as $$ \begin{aligned} \tilde{R}_{x_1,x_2}(l) =& \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{R_{x_1,x_2}(\omega)}{|R_{x_1,x_2}(\omega)|}e^{j\omega l}d\omega \\ =& \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{j\omega(l + \tau_0)} d\omega \\ &= \delta(l - \tau_0) \end{aligned} $$ where $\delta$ is the Dirac-delta function which is non-zero at $l = \tau_0$, which is the time-difference of arrival (TDOA). Typically, the lag is calculated in samples, so to convert that to seconds, you have to divide it by the sampling rate.

I hope this helps in understanding GCC-PHAT, which is useful tool for TDOA estimation, and hence, source localization.

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orchi_d
orchi_d
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Let there be two microphones, $x_1(t), x_2(t)$ capturing a source, $s(t)$, with some additive noise $n_{1,2}(t)$ that is uncorrelated with the source and with each other (i.e., its effect can be ignored while calculating the cross-correlation). Let $\tau_0$ (in seconds) be the time difference of arrival between the two microphones. The time domain equations of the microphones are $$x_1(t) = s(t) + n_1(t) \\ x_2(t) = \alpha_0 s(t - \tau_0) + n_2(t) $$

Now, taking the FFT and converting these to the frequency domain would give us, $$ X_1(\omega) = S(\omega) + N_1(\omega) \\ X_2(\omega) = \alpha_0 S(\omega) \exp{(-j\omega \tau_0)} + N_2(t) $$

The cross correlation function of the two microphone signals (or the cross-power spectrum in the frequency domain), can be written as

$$ \begin{aligned} R_{x_1,x_2}(l) &= \mathbb{E}[x_1(t) x_2(t-l)] \\ R_{x_1, x_2}(\omega) &= X_1(\omega)X_2^*(\omega) \\ &= |S(\omega)|^2 \alpha_0 \exp{(j\omega \tau_0)} \end{aligned} $$

The Generalized Cross-Correlation function with Phase Transform (GCC-PHAT) at lag $l$ is defined as $$ \begin{aligned} \tilde{R}_{x_1,x_2}(l) =& \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{R_{x_1,x_2}(\omega)}{|R_{x_1,x_2}(\omega)|}e^{j\omega l}d\omega \\ =& \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{j\omega(l + \tau_0)} d\omega \\ &= \delta(l - \tau_0) \end{aligned} $$ where $\delta$ is the Dirac-delta function which is non-zero at $l = \tau_0$, which is the time-difference of arrival (TDOA). Typically, the lag is calculated in samples, so to convert that to seconds, you have to divide it by the sampling rate.

I hope this helps in understanding GCC-PHAT, which is useful tool for TDOA estimation, and hence, source localization.