Calculate bandwidth of a signal for Nyquist–Shannon sampling theorem

I have to calculate the minimum sampling frequency for the Nyquist–Shannon sampling theorem which is Fc > 2*B, where B is the signal bandwidth. I have this signals: $$\text{sinc}^5(t/2 - 4)$$ and $$\text{sinc}^3(3 - 2t)$$

How can I calculate the signal bandwidth to obtain the minimum sampling frequency for the Nyquist–Shannon sampling theorem?

• Hi! Are you a student ? is this a homework? please indicate. – Fat32 Jan 21 at 16:44
• what do mean by sinc^(3-2t)? Is it $\text{sinc}(3-2t)$? – BlackMath Jan 21 at 16:49
• @BlackMath I’ve corrected the function. Is sinc^3(3 - 2t) – Peter Vogric Jan 21 at 16:55
• @Fat32 Yes I’m a student and this is a quesiton of my test at university – Peter Vogric Jan 21 at 16:56
• ok. are you taking a signals course ? – Fat32 Jan 21 at 16:56

1 Answer

So apply the following principals to get to a solution:

1. Shifts in the time domain don't affect the bandwidth, so the $$-4$$ and the $$+3$$ can be ignored.
2. Reversal of the time domain axis doesn't affect the bandwidth, so you can ignore the negative sign on the $$-2t$$.
3. The $$\mathrm{sinc}()$$ function has a Fourier transform that is a rectangle function of a particular width.
4. Scaling the time axis (lengthening or contracting) of a function, scales the frequency axis in the opposite manner (contracting or lengthening) for the Fourier transform.
5. Multiplication of functions in the time domain, results in the convolution of those two functions' Fourier transforms in the frequency domain.
6. The convolution of two Fourier transforms in the frequency domain, will result in a Fourier transform whose bandwidth is the sum of the bandwidth of those initial two Fourier Transforms.

That should give you enough to come up with an answer.

• I still can’t get it. Can you answer one of those examples. I’ve understood the theory behind that, but I’m not sure about my answer. – Peter Vogric Jan 22 at 9:22