# Impulse response of a 3x3 PSF - how to find analytical expression for fourier transform of a 3x3 matrix?

I have a filter $$\mu[n_1, n_2]$$ with taps:

$$(1/8) (1/4) (1/8)$$ $$(1/4) (1/2) (1/4)$$ $$(1/8) (1/4) (1/8)$$

How do I find an analytical expression for $$\hat\mu(w_1, w_2)$$?

Since it looks so much like a triangle, I feel like it looks something of the form $$sinc^2(w)$$, but the exact answer I need help with.

• You can try to simplicity the problem by noticing that the filter is separable, and you can start with 1D triangle, then combine them into 2D – Laurent Duval Aug 22 at 17:05
• @LaurentDuval So I know that the filter is the product of two triangle filters $\mu(n_1) = (1, 2, 1) = 2Tri(2n_2)$ (?) and $\mu(n_2) = (1/8, 1/4, 1/8)^T = 1/4tri(2n_2)$ (?). Does this mean that $\hat\mu(w_1) = 2 \pi sinc^2(w_1/2)$ and $\hat\mu(w_2) = \pi/2 sinc^2(w_2/2)$ So – Nobody Special Aug 23 at 3:55
• So then $\hat \mu(w_1,w_2) = \mu(w_1) \cdot \mu(w_2)$ ?@Fat32 – Nobody Special Aug 23 at 4:02
• Why not try 2d FFT of the filter? – Moti Aug 23 at 5:32

You can use the 2D DFT formula

$$H(\omega_1,\omega_2) = \sum_{n_1} \sum_{n_2} h[n_1,n_2] e^{-j(\omega_1 n_1 + \omega_2 n_2)}$$

and simply the trigonometric algebra to get a closed form analytic expression for the 2D-DTFT. However, as @LaurentDuval has already mentioned, your 3x3 kernel is separable and one set of 1D filters is this

$$f[n_1] = [\frac{1}{4} , \frac{1}{2}, \frac{1}{4}]^T$$ $$g[n_2] = [\frac{1}{2} , 1 , \frac{1}{2}]$$

Then from the Fourier transform we know that

$$h[n_1,n_2] = f[n_1]g[n_2] \implies H(\omega_1,\omega_2) = F(\omega_1) G(\omega_2)$$

Analytic expressions for 1D-DTFT's can be obtained from $$H(\omega) = \sum_{n}h[n] e^{-j \omega n}$$

Assuming 3x3 kernel has zero phase, then we see that : $$F(\omega_1) = \frac{1}{4} e^{j \omega_1 } + \frac{1}{2} e^{j 0 } + \frac{1}{4} e^{-j \omega_1 } = 0.5 \cos(\omega_1) + 0.5$$

similarly $$G(\omega_2) = \frac{1}{2} e^{j \omega_2 } + 1 e^{j 0 } + \frac{1}{2} e^{-j \omega_2 } = \cos(\omega_2) + 1$$

Hence: $$H(\omega_1,\omega_2) = 0.5 \left( \cos(\omega_1) + 1 \right)(\cos(\omega_2) + 1)$$

• why do you assume the kernel is zero phase? what does that do? – Nobody Special Aug 25 at 11:16
• Just for simplicity, and correct unless otherwise stated... You can consider non-zero phase as well. You will just multiply the frequency response with a complex exponential linear phase term... – Fat32 Aug 25 at 11:33