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Determine the autocorrelation $r_{xx}[m]$ of the discrete signal

$$x[n] = (\sin{2\pi fn)}.$$

where $n$ and $m$ are integers.

Using the definition I get

$$\begin{align} r_{xx}[m] &= \sum_{n=-\infty}^{\infty}x[n]x[n-m] \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi f(n-m))}} \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi fn - 2\pi fm))}} \\ \end{align}$$

but I can't seem to figure it out from here. I've tried using different trigonometric identities without result. I'm guessing it's something simple I'm missing.

Determine the autocorrelation $r_{xx}[m]$ of the discrete signal

$$x[n] = (\sin2\pi fn).$$

where $n$ and $m$ are integers.

Using the definition I get

$$\begin{align} r_{xx}[m] &= \sum_{n=-\infty}^{\infty}x[n]x[n-m] \\ &= \sum_{n=-\infty}^{\infty}\sin(2\pi fn) \sin\big(2\pi f(n-m)\big) \\ &= \sum_{n=-\infty}^{\infty}\sin(2\pi fn) \sin\big(2\pi fn - 2\pi fm\big) \\ \end{align}$$

but I can't seem to figure it out from here. I've tried using different trigonometric identities without result. I'm guessing it's something simple I'm missing.

Determine the autocorrelation $r_{xx}[m]$ of the discrete signal

$$x[n] = (\sin{2\pi fn)}.$$

where $n$ and $m$ are integers.

Using the definition I get

$$\begin{align} r_{xx}[m] &= \sum_{n=-\infty}^{\infty}x[n]x[n-m] \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi f(n-m))}} \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi fn - 2\pi fm))}} \\ \end{align}$$

but I can't seem to figure it out from here. I've tried using different trigonometric identities without result. I'm guessing it's something simple I'm missing.

Determine the autocorrelation $r_{xx}[m]$ of the discrete signal

$$x[n] = (\sin{2\pi fn)}.$$

where $n$ and $m$ are integers.

Using the definition I get

$$\begin{align} r_{xx}[m] &= \sum_{n=-\infty}^{\infty}x[n]x[n-m] \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi f(n-m))}} \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi fn - 2\pi fm))}} \\ \end{align}$$

but I can't seem to figure it out from here. I've tried using different trigonometric identities without result. I'm guessing it's something simple I'm missing.

Determine the autocorrelation $r_{xx}[m]$ of the discrete signal

$$x[n] = (\sin{2\pi fn)}.$$

where $n$ and $m$ are integers.

Using the definition I get

$$\begin{align} r_{xx}[m] &= \sum_{n=-\infty}^{\infty}x(n)x(n-m) \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi f(n-m))}} \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi fn - 2\pi fm))}} \\ \end{align}$$

but I can't seem to figure it out from here. I've tried using different trigonometric identities without result. I'm guessing it's something simple I'm missing.

Determine the autocorrelation $r_{xx}[m]$ of the discrete signal

$$x[n] = (\sin{2\pi fn)}.$$

where $n$ and $m$ are integers.

Using the definition I get

$$\begin{align} r_{xx}[m] &= \sum_{n=-\infty}^{\infty}x[n]x[n-m] \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi f(n-m))}} \\ &= \sum_{n=-\infty}^{\infty}\sin{(2\pi fn)\sin{(2\pi fn - 2\pi fm))}} \\ \end{align}$$

but I can't seem to figure it out from here. I've tried using different trigonometric identities without result. I'm guessing it's something simple I'm missing.