# How to change a continuous signal to form a discrete signal? [closed]

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 $x[n]$ ?

• I don't understand your question; making a discrete signal out of a continuous one is exactly what sampling is. – Marcus Müller Oct 24 '16 at 15:54

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.