AN2182

Low-pass filter

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Figure 3.

Ideal low-pass filter impulse response

Two issues can be observed at this stage:

First, the filter length is infinite. This means that in order to have the ideal low-pass filter, an

infinite number of filter coefficients is required. In practice, this cannot be done because of the

calculation complexity. To solve this feasibility problem, the filter’s impulse response should be

truncated by a known window. In this application note, we will truncate this response using a

rectangular window.

Truncating the filter’s impulse response, however, changes the ideal rectangular response;

more fluctuations are observed in the pass band. In order to reduce this effect, the FIR’s

coefficients’ numbers can be increased.

Figure 4.

Effect of truncating the ideal impulse with a rectangular window

In the following sections, we will note:

●

W is the window’s time response

●

N the number of coefficients included in the window W. This means that the window’s

the sinus cardinal.

Second, the truncated response is finite but not causal because there are non-null coefficients

on the negative time axis. This means that the filter reacts before being excited by the input

signal. Practically, this filter is not feasible. To solve this issue, we will shift the filter’s impulse

becomes feasible.

The impact of shifting the filter’s time response is a phase shift of

in the

frequency domain.

k>=1

0

1

Frequency

Frequency domain

Time domain

Time

W(n)

N1

–

() π

f

F

s

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