S5-3.3 FIR Filter Design & Realization

What this is and why it exists

A filter delays different frequencies by different amounts unless it is built not to. When a signal carries pulses, that uneven delay smears them.

Finite impulse response filters can have exactly linear phase, which means every frequency is delayed equally. A pulse comes out later but unchanged in shape. That is the whole reason this family exists, and it matters most in data transmission and imaging.

The vocabulary

• FIR filter — a filter whose output depends only on a finite window of past inputs.
• Linear phase — a phase response that is a straight line, so all frequencies are delayed equally.
• Symmetry — coefficients that read the same forwards and backwards, or the same but negated.
• Main lobe — the central peak of a window's frequency response. Its width sets the transition width.
• Side lobes — the smaller peaks beside it. Their height limits the stopband attenuation.
• Rectangular window — cutting the ideal response off abruptly.
• Bartlett, Hamming and Blackman windows — tapered windows, each with a different compromise.
• Kaiser window — a window with a parameter that lets you set the compromise yourself.
• Frequency sampling — designing by specifying the response at points and transforming back.
• Linear phase structure — a realization that exploits symmetry to halve the multipliers.

The mental model

Linear phase comes from symmetry in the coefficients, and nothing else. That symmetry can be even or odd, and the filter length can be even or odd, which gives four types. The classification is not bookkeeping. Some types force a zero in the response at zero frequency or at half the sampling rate. Knowing which is which stops you designing something that cannot exist. A highpass filter, for instance, rules out the type whose response must vanish at half the sampling rate.

Windowing is the main design route, and it has one honest story. Start from the ideal response you want. Its impulse response is infinitely long, so it must be cut off. Cutting it off is multiplying it by a window.

Every window is a point on one trade. The rectangular window cuts abruptly. Its main lobe is the narrowest of all, so the transition from pass to stop is the sharpest. Its side lobes are the worst, so the stopband leaks badly. Taper the window and the side lobes drop, but the main lobe widens and the transition softens. Bartlett, Hamming and Blackman are three fixed points along that line, in roughly that order.

Kaiser is the one to reach for once you understand the others. It has a parameter that moves it along the trade. You can meet a stated stopband attenuation and transition width, rather than trying windows until one fits.

Frequency sampling is a different route. Instead of starting from an ideal shape, state the response you want at a set of evenly spaced frequencies and transform back. It is useful when the shape you want is not one of the standard four.

Realization repays the symmetry. In a direct form structure every coefficient needs its own multiplier. In the linear phase structure the symmetric pairs are added before multiplying, so about half the multipliers disappear. The response is identical. That is mathematics buying hardware, and on a real processor the saved multiplies are cycles you get back.

What you should now be able to explain or do

State why linear phase matters and where it comes from. Name the four types and say which responses each cannot realise. Explain the window trade in terms of main lobe width and side lobe height. Choose a window for a stated transition width and stopband attenuation, and say why Kaiser is the flexible choice. Draw the linear phase structure and explain where the halved multiplier count comes from.

Check yourself