Topic 3: FIR Filter Design

Implementation

An FIR digital filter can be written as an LLCD equation

Ideal Frequency Filters

Ideal but are non causal and infinite length so are not practical

Can be implemented by shifting their response then truncating the impulse response

Window Design Method

Given passband and stopband cutoff frequencies, a sample frequency and a stopband attenuation factor:

1. Convert frequencies to normalised units

2. Convert to radians/sample

3. Select appropriate window function with stopband attenuation factor

4. Determine number of points N required in the window to meet the spec (N should be odd so +1 if even)

   1. Transition bandwidth = stopband (rad/sample) - passband (rad/sample) = Main lobe width from window function

5. Final filter multiplies sinc by selected window function:

   1. h[n] = sinc [ w_pass * (n - (N - 1) / 2) ] * W [ (n - (N - 1) / 2) ]

Frequency Sampling Method

Given a passband cutoff frequency, a sample frequency, and a filter order:

• Calculate the normalised cutoff frequency

• Convert to radians/sample

• Draw ideal frequency response with cutoff frequency

• Place sample at intervals of: 2 * pi / N (use odd N to be centred)

• Calculate inverse DFT

Converting to a High Pass Design

Apply a frequency shift with phase ramp to the filter

h_HP[n] = e^-jwn h_LP[n]

Parallel Filters

OR gate

Bandstop (notch) filter

Cascade Filters

AND gate

Bandpass filter