# Can I use a transfer function to filter noise?

I want to make it simple! I want to filter white noise with a transfer function and it's going to be zero phase.

Assume that we have our low pass filter.

$$G(s) = \frac {1}{1 + Ts}$$

Where $$T$$ is my tuning parameter.

And then www.deviantart.com apply our noisy data $$u(t)$$ to get our filtered output $$y(t)$$

$$y(t) = G(s)u(t)$$

Then i flipp $$y(t)$$ to $$y(-t)$$ And do the same process again.

$$u(-t) = G(s)y(-t)$$

And now i flipp $$u(-t)$$ to $$u(t)$$.

Questions:

1. Is Discrete Fourier Transform better to use instead of a low pass filter?
2. Will this method work? Using transfer functions.

Thank you.

DFT is just a tool to convert time domain samples to frequency domain. To filter your discrete data, you can just perform DFT on the input data, multiply it with the transfer function of the LPF and then take the inverse DFT.

Q1: Discrete Fourier Transform is not a filtering method. The comparison is wield.

Q2: Because you didn't provide any information about your spec, applications, no one can tell you the answer. Filter design is a kind of design. There is no absolute answer. It depends!

What I'm going to do is show more physical meanings about the low pass filter you mentioned. Then maybe you can figure out whether the filter is what you want.

Because you mentioned Discrete Fourier Transform, I guess that you're actually dealing with discrete time signals instead of continuous ones. So I rewrite the discrete form low pass filter (LPF) as follow and simulating them accordingly. The other reason is that discrete form is much easier to simulate via code.

The LPF you mentioned is actually a kind of exponential smoothing.

Difference Equation: $$y[n] = \alpha x[n] + (1 - \alpha) y[n-1]$$ This form is very simple (low complexity), intuitive. So it's commonly utilized in some applications. But relative few designers can interpret the physical meanings.

Z-transform: (The $$1- \alpha$$ in numerator is designed for unit DC gain shown after.) $$H[z]=\frac{1 - \alpha}{1 - \alpha z^{-1}},\quad 0 \leq k \leq N-1$$

BTW, $$|\alpha| < 1$$ most be held for stability.

Impulse response (It's a kind of IIR filter.): $$h[n]= (1-\alpha )\alpha^{n- 1}.$$

DC-gain: $$H[z ]|_{z=exp(j0)=1}= \frac{1 - \alpha}{1 - \alpha} = 1$$

Max Attenu: $$H[z ]|_{z=exp(j \pi)=-1}= \frac{1 - \alpha}{1 + \alpha }$$

Freq. Response:

You can also tune the $$\alpha$$ via the concept of time constant. The time constant of an exponential moving average is the amount of time for the smoothed response of a unit set function to reach $$1-\frac{1}{e}=63.2\%$$. Given time constant ,$$\tau$$, to tune the $$\alpha$$: $$\alpha = 1 - e^{\frac{-T}{\tau}},$$ where T is sampling time.

Code:

alpha = 0.1;

b = [1 alpha];
a = [1 -alpha];

fvtool(b,a);

In addition, the flipping processing is the so-called filpflip via which you can obtain zero phase delay at the expanse of causality.