Discrete time signal to continuous time signal

[Lightning19]

Homework Statement

Suppose that a discrete-time signal x[n] is given by the formula

x[n] = 10cos(0.2*PI*n - PI/7)

and that it was obtained by sampling a continuous signal at a sampling rate of fs=1000 samples/second.

Determine two different continuous-time signals x1(t) and x2(t) whose samples are equal to x[n]; ie. find x1(t) and x2(t) such that x[n] = x1(nTs) = x2(nTs0 if Ts = 0.001. Both of these signals should have a frequency less than 1000Hz. Give a formula for each signal.

The Attempt at a Solution

Since x[n] = 10 * cos ( 0.2 * PI * n - PI / 7)
The same signal would be represented by

x[n] = 10 * cos(2.2 * PI * n - PI/7)

Is this assumption correct?

Now I would need to convert these two x[n] equations to time-domain x(t), and I am not sure how I would go about doing this part.

 

[Valkarie]

Could someone please provide some guidance on how I can go about doing this? Thank you very much in advance.

 

[piercas]

Can you please give me a hint or guide me in the right direction?

Your assumption is not correct. The two signals x[n] and x1[n] have the same samples but they are not the same signal. To find the continuous-time signal corresponding to x[n], we can use the formula x(t) = x[n]*sinc(π(t/Ts - n)), where Ts is the sampling period and sinc(x) = sin(πx)/(πx).

Therefore, the two continuous-time signals corresponding to x[n] can be written as:

x1(t) = 10 * cos(0.2 * π * t - π/7) * sinc(π(t/0.001 - n))
x2(t) = 10 * cos(2.2 * π * t - π/7) * sinc(π(t/0.001 - n))

Both of these signals have a frequency of 0.2 Hz, which is less than the Nyquist frequency of 500 Hz (fs/2 = 1000/2 = 500 Hz).

To understand how these signals were derived, we can think of the continuous-time signal as being composed of an infinite number of samples, each multiplied by a sinc function. The sinc function acts as a low-pass filter, allowing only the frequency components within the Nyquist frequency to pass through. Therefore, the two signals x1(t) and x2(t) are essentially low-pass filtered versions of x[n].