- #1
Master1022
- 611
- 117
- Homework Statement
- We have a time-continuous signal [itex] f(t) [/itex]. A new signal [itex] g(t) [/itex] is created by either by multiplying [itex] f(t) [/itex] with a top-hat function (half-width [itex] \frac{T}{2} [/itex]) or a ramp function (half-width [itex] T [/itex]), both with amplitude 1. Which window should we, using qualitative judgement, choose to have a better representation of [itex] F(\omega ) [/itex]
- Relevant Equations
- Fourier transform
Hi,
I was hoping to gain more insight into these window questions when looking at frequency spectra questions. I don't really know what makes windows better than one another.
My attempt:
In the question, we have [itex] f(t) = cos(\omega_0 t) [/itex] and therefore its F.T is [itex] F(\omega ) = \pi \left( \delta(\omega - \omega_0 ) + \delta(\omega + \omega_0) \right) [/itex]. For the window functions, we have a top-hat function with a transform of:
[tex] \frac{ T sin(\omega T / 2)}{\omega T / 2} [/tex] and a ramp function with transform:
[tex] \frac{ 4 sin^2 (\omega T / 2)}{\omega^2 T} [/tex]
To find the effect of multiplying the time signals, we can carry out convolution in the time domain and utilize the sifting property of the delta function.
I can see that we basically have the choice of [itex] sinc [/itex] or [itex] sinc^2 [/itex]. Perhaps the ramp function will be better as it has smaller peripheral pulses (due to the sinc function being squared). Also, I notice that the ramp function is the result of convolving the top-hat function with itself.
I am not sure what other aspects I should be looking out for.
Any help is greatly appreciated.
I was hoping to gain more insight into these window questions when looking at frequency spectra questions. I don't really know what makes windows better than one another.
My attempt:
In the question, we have [itex] f(t) = cos(\omega_0 t) [/itex] and therefore its F.T is [itex] F(\omega ) = \pi \left( \delta(\omega - \omega_0 ) + \delta(\omega + \omega_0) \right) [/itex]. For the window functions, we have a top-hat function with a transform of:
[tex] \frac{ T sin(\omega T / 2)}{\omega T / 2} [/tex] and a ramp function with transform:
[tex] \frac{ 4 sin^2 (\omega T / 2)}{\omega^2 T} [/tex]
To find the effect of multiplying the time signals, we can carry out convolution in the time domain and utilize the sifting property of the delta function.
I can see that we basically have the choice of [itex] sinc [/itex] or [itex] sinc^2 [/itex]. Perhaps the ramp function will be better as it has smaller peripheral pulses (due to the sinc function being squared). Also, I notice that the ramp function is the result of convolving the top-hat function with itself.
I am not sure what other aspects I should be looking out for.
Any help is greatly appreciated.