# Fourier transform of the product

I'm new at signal processing. I also don't know if this is the right section to post. I have this problem: I must calculate the Fourier transform of the signal shown in the image below I have expressed this as $$\require{cancel}\cancel{\Delta\left(\frac{t-4}{4}\right)\textrm{rect}\left(\frac{t-2}{4}\right)}$$

I read that the Fourier transform of a time-domain product equals the convolution of the two transforms in frequency domain but I don't know how to proceed, can you kindly help me?

• I suggest you first plot the signal in time domain. Then, you find a simpler form to describe it in time-domain, and then you evaluate the Fourier-Integral directly. Dont try the convolution in frequency domain, as you would need to convolve sinc^2 with sinc, which can be quite difficult. – Maximilian Matthé Jan 5 '17 at 10:06
• This originally was a graph, and I wrote the signal. I don't know a simpler way. – lucad93 Jan 5 '17 at 10:10
• So, how does the graph look like? – Maximilian Matthé Jan 5 '17 at 10:14
• It's like a scalene trapezoid, I try to put a photo into the post. – lucad93 Jan 5 '17 at 10:20
• Calculate the FT of derivative of this signal (which is the difference of two sinc functions). Then use the properties of FT. – msm Jan 5 '17 at 10:33

First of all, your graph does not match the equation you gave for the function. From the hint of msm, we should use the following property of the Fourier Transform:

$$\mathcal{F}\left\{\frac{d}{dt}x(t)\right\}=j2\pi fX(f)$$

with

$$\mathcal{F}\{x(t)\} = X(f)$$

Now, what's the derivative of your function in the graph? Its given by

$$x'(t)=\frac{1}{4}\text{rect}\left(\frac{t-2}{4}\right) - \text{rect}(t-\frac{9}{2})$$

Now, let's do the Fourier Transform of the derivative:

$$j2\pi fX(f)=\text{sinc}(4f)\exp(-j2\pi 2f) - \text{sinc}(f)\exp(-j2\pi \frac{9}{2}f)$$

with $\text{sinc}(x)=\frac{\sin(\pi x)}{\pi x}$.

Now, finally dividing by $j2\pi f$ you get

$$X(f) = \frac{1}{j2\pi f}\left(\text{sinc}(4f)\exp(-j2\pi2f)-\text{sinc}(f)\exp(-j2\pi\frac{9}{2}f)\right).$$

You can try to simplify this expression, but it might not yield something that is better to understand.

It's much easier to approach this particular function differently: differentiate it and you get two rectangles which are comparatively easy to Fourier transform. Integration is convolution with a unit step function, so you can then multiply the transform of the rectangle sequence with the transform of the unit step function (depends on the Fourier transform convention used) in order to retrieve the original function.