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I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT.

For example if I take a basic example:

$$\begin{aligned} x(t) &= cos(\omega_x \cdot t) = \frac{1}{2} \cdot \left( e^{j\omega_x t} + e^{-j\omega_x t} \right) \\ \implies X(\omega) &= \pi \cdot (\delta(\omega - \omega_x) + \delta(\omega+\omega_x)) \\ x_c(t) &= x(t) \cdot \sum_{n=-\infty}^{\infty}\delta(t - nT_s) = \sum_{n=-\infty}^{\infty}x(nT_s)\delta(t-nT_s) \\ \implies X_c(\omega) &= X(\omega) * \left( \omega_s \sum_{n=-\infty}^{\infty} \delta(\omega - n\omega_s)\right) \\ \end{aligned}$$

$$ X_c(\omega) = \omega_s \pi \sum_{n=-\infty}^{\infty} \left( \delta(\omega - \omega_x - n\omega_s) + \delta(\omega + \omega_x - n\omega_s) \right) \tag{1} \label{1} $$

From there I'm lost and everything crumbles. I'm only trying to get the DTFT of a cosine which is: $$ cos(\Omega_0 n) \Longleftrightarrow \pi \sum_{n=-\infty}^{\infty} \left( \delta(\Omega - \Omega_0 - n2\pi) + \delta(\Omega + \Omega_0 - n2\pi) \right) \tag{2} \label{2} $$

How can I obtain $\eqref{2}$ starting from $\eqref{1}$?

I hope what I'm trying even makes sense. After all the DTFT with infinite period is the CTFT so I suppose there's a link we can make between these equations?

Thanks

I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT.

For example if I take a basic example:

$$\begin{aligned} x(t) &= \cos(\omega_x \cdot t) = \frac{1}{2} \cdot \left( e^{j\omega_x t} + e^{-j\omega_x t} \right) \\ \implies X(\omega) &= \pi \cdot (\delta(\omega - \omega_x) + \delta(\omega+\omega_x)) \\ x_c(t) &= x(t) \cdot \sum_{n=-\infty}^{\infty}\delta(t - nT_s) = \sum_{n=-\infty}^{\infty}x(nT_s)\delta(t-nT_s) \\ \implies X_c(\omega) &= X(\omega) * \left( \omega_s \sum_{n=-\infty}^{\infty} \delta(\omega - n\omega_s)\right) \\ \end{aligned}$$

$$ X_c(\omega) = \omega_s \pi \sum_{n=-\infty}^{\infty} \left( \delta(\omega - \omega_x - n\omega_s) + \delta(\omega + \omega_x - n\omega_s) \right) \tag{1} \label{1} $$

From there I'm lost and everything crumbles. I'm only trying to get the DTFT of a cosine which is: $$ \cos(\Omega_0 n) \Longleftrightarrow \pi \sum_{n=-\infty}^{\infty} \left( \delta(\Omega - \Omega_0 - n2\pi) + \delta(\Omega + \Omega_0 - n2\pi) \right) \tag{2} \label{2} $$

How can I obtain $\eqref{2}$ starting from $\eqref{1}$?

I hope what I'm trying even makes sense. After all the DTFT with infinite period is the CTFT so I suppose there's a link we can make between these equations?

Thanks

I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT.

For example if I take a basic example: $$ x(t) = cos(\omega_x \cdot t) = \frac{1}{2} \cdot \left( e^{j\omega_x t} + e^{-j\omega_x t} \right) $$ $$ \implies X(\omega) = \pi \cdot (\delta(\omega - \omega_x) + \delta(\omega+\omega_x)) $$ $$ x_c(t) = x(t) \cdot \sum_{n=-\infty}^{\infty}\delta(t - nT_s) = \sum_{n=-\infty}^{\infty}x(nT_s)\delta(t-nT_s) $$

$$ \implies X_c(\omega) = X(\omega) * \left( \omega_s \sum_{n=-\infty}^{\infty} \delta(\omega - n\omega_s)\right) $$ $$ X_c(\omega) = \omega_s \pi \sum_{n=-\infty}^{\infty} \left( \delta(\omega - \omega_x - n\omega_s) + \delta(\omega + \omega_x - n\omega_s) \right) \tag{1} \label{1} $$

From there I'm lost and everything crumbles. I'm only trying to get the DTFT of a cosine which is: $$ cos(\Omega_0 n) \Leftrightarrow \pi \sum_{n=-\infty}^{\infty} \left( \delta(\Omega - \Omega_0 - n2\pi) + \delta(\Omega + \Omega_0 - n2\pi) \right) \tag{2} \label{2} $$

How can I obtain $\eqref{2}$ starting from $\eqref{1}$? I hope what I'm trying even makes sense. After all the DTFT with infinite period is the CTFT so I suppose there's a link we can make between these equations?

Thanks

I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT.

For example if I take a basic example:

$$\begin{aligned} x(t) &= cos(\omega_x \cdot t) = \frac{1}{2} \cdot \left( e^{j\omega_x t} + e^{-j\omega_x t} \right) \\ \implies X(\omega) &= \pi \cdot (\delta(\omega - \omega_x) + \delta(\omega+\omega_x)) \\ x_c(t) &= x(t) \cdot \sum_{n=-\infty}^{\infty}\delta(t - nT_s) = \sum_{n=-\infty}^{\infty}x(nT_s)\delta(t-nT_s) \\ \implies X_c(\omega) &= X(\omega) * \left( \omega_s \sum_{n=-\infty}^{\infty} \delta(\omega - n\omega_s)\right) \\ \end{aligned}$$

$$ X_c(\omega) = \omega_s \pi \sum_{n=-\infty}^{\infty} \left( \delta(\omega - \omega_x - n\omega_s) + \delta(\omega + \omega_x - n\omega_s) \right) \tag{1} \label{1} $$

From there I'm lost and everything crumbles. I'm only trying to get the DTFT of a cosine which is: $$ cos(\Omega_0 n) \Longleftrightarrow \pi \sum_{n=-\infty}^{\infty} \left( \delta(\Omega - \Omega_0 - n2\pi) + \delta(\Omega + \Omega_0 - n2\pi) \right) \tag{2} \label{2} $$

How can I obtain $\eqref{2}$ starting from $\eqref{1}$?

I hope what I'm trying even makes sense. After all the DTFT with infinite period is the CTFT so I suppose there's a link we can make between these equations?

Thanks

I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT.

For example if I take a basic example: $$ x(t) = cos(\omega_x \cdot t) = \frac{1}{2} \cdot \left( e^{j\omega_x t} + e^{j\omega_x t} \right) $$ $$ \implies X(\omega) = \pi \cdot (\delta(\omega - \omega_x) + \delta(\omega+\omega_x)) $$ $$ x_c(t) = x(t) \cdot \sum_{n=-\infty}^{\infty}\delta(t - nT_s) = \sum_{n=-\infty}^{\infty}x(nT_s)\delta(t-nT_s) $$

$$ \implies X_c(\omega) = X(\omega) * \left( \omega_s \sum_{n=-\infty}^{\infty} \delta(\omega - n\omega_s)\right) $$ $$ X_c(\omega) = \omega_s \pi \sum_{n=-\infty}^{\infty} \left( \delta(\omega - \omega_x - n\omega_s) + \delta(\omega + \omega_x - n\omega_s) \right) \tag{1} \label{1} $$

From there I'm lost and everything crumbles. I'm only trying to get the DTFT of a cosine which is: $$ cos(\Omega_0 n) \Leftrightarrow \pi \sum_{n=-\infty}^{\infty} \left( \delta(\Omega - \Omega_0 - n2\pi) + \delta(\Omega + \Omega_0 - n2\pi) \right) \tag{2} \label{2} $$

How can I obtain $\eqref{2}$ starting from $\eqref{1}$? I hope what I'm trying even makes sense. After all the DTFT with infinite period is the CTFT so I suppose there's a link we can make between these equations?

Thanks

I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT.

For example if I take a basic example: $$ x(t) = cos(\omega_x \cdot t) = \frac{1}{2} \cdot \left( e^{j\omega_x t} + e^{-j\omega_x t} \right) $$ $$ \implies X(\omega) = \pi \cdot (\delta(\omega - \omega_x) + \delta(\omega+\omega_x)) $$ $$ x_c(t) = x(t) \cdot \sum_{n=-\infty}^{\infty}\delta(t - nT_s) = \sum_{n=-\infty}^{\infty}x(nT_s)\delta(t-nT_s) $$

$$ \implies X_c(\omega) = X(\omega) * \left( \omega_s \sum_{n=-\infty}^{\infty} \delta(\omega - n\omega_s)\right) $$ $$ X_c(\omega) = \omega_s \pi \sum_{n=-\infty}^{\infty} \left( \delta(\omega - \omega_x - n\omega_s) + \delta(\omega + \omega_x - n\omega_s) \right) \tag{1} \label{1} $$

From there I'm lost and everything crumbles. I'm only trying to get the DTFT of a cosine which is: $$ cos(\Omega_0 n) \Leftrightarrow \pi \sum_{n=-\infty}^{\infty} \left( \delta(\Omega - \Omega_0 - n2\pi) + \delta(\Omega + \Omega_0 - n2\pi) \right) \tag{2} \label{2} $$

How can I obtain $\eqref{2}$ starting from $\eqref{1}$? I hope what I'm trying even makes sense. After all the DTFT with infinite period is the CTFT so I suppose there's a link we can make between these equations?

Thanks