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I have a continuous signal $$ x(t)= 2\cos(200\pi t)+8\cos(250\pi t)$$ and it is sampled at 400Hz.

How is this changed into the discrete time domain in order to determine the two discrete frequency components of $x[n]$ ?

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  • $\begingroup$ I don't understand your question; making a discrete signal out of a continuous one is exactly what sampling is. $\endgroup$ – Marcus Müller Oct 24 '16 at 15:54
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When your continuous-time signal $$x(t)= 2\cos(200\pi t)+8\cos(250\pi t)$$ is sampled at a rate of $F_s = 400$ Hz, (or sampled with a period of $T_s= \frac{1}{F_s} = \frac{1}{400}$ seconds per sample), you can use the following to find the mathematical representation of the discrete-time signal $x[n]$ :

$$ x[n] = x_c(t)|_{t=nT_s} $$

Applying this to $x(t)$ leads to: $$x[n] = x(nT_s)= 2\cos(200\pi n T_s)+8\cos(250\pi n T_s)$$ $$x[n] = x(n/400)= 2\cos(200 \pi n /400)+8\cos(250\pi n /400)$$ $$x[n] = 2 \cos(\frac {2\pi}{4} n) + 8 \cos(\frac {5\pi}{8} n)$$

Now observing the pattern of sinusoidal signals in discrete time as $\cos(\omega_0 n)$ where $\omega_0$ is the discrete time radian frequency between $-\pi$ and $\pi$ (or $0$ to $2\pi$ if you prefer so) you can deduce your discrete time frequencies as $\pi/2$ and $5\pi/8$

Note that $\omega_0 = \pi$ will be the highest frequency discrete time sinusoid.

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