Digital filtering principles

AN2182

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Digital filtering principles

Assume a continuous signal x(t) (the complex form corresponds to the signal’s phase and

magnitude at the instant t) with a pass band B. Assume that this signal will be filtered using a

filter with a continuous impulse response h(t).

When digital processing has to be used, it is necessary to sample the input signal with a

from the samples obtained at the filter’s output.

Figure 1.

Example of input and output signals

The Shannon theorem states that when sampling a signal at discrete intervals, the sampling

2.1

Fourier transform of a sampled signal

Signals are converted from the time domain to the frequency domain usually through the

Fourier transform. With the Fourier transform, the signal is converted to a magnitude and phase

at each frequency.

The Fourier transform of the sampled signal x(k) has the following expression:

The time representation can be computed from the X(f) as follows:

2.2

Linear filtering

Using the notations defined in the previous section, the output signal y(n) is the convolution of

the input signal x(k) and the filter impulse response h(k) .

Filter h

Input signal

Output signal

Xf

()

xk

()e

i2k

πf

∞

–

∞

∑

=

xk

()

Xf

()e

i

– 2k

πf

∞

–

∞

∑

=

yn

()

hn

k

–

()xk

()

∞

–

∞

∑

hk

() xk

()

⊗

()

n

==