FIR and IIR filters

Conventions used throughout. Digital frequency $\omega$ is in rad/sample and analog frequency $\Omega$ is in rad/s. When a digital filter processes samples taken every $T=1/f_s$ seconds, an input component at $f$ Hz ($\Omega =2\pi f$ rad/s) appears at $\omega =\Omega T=2\pi f/f_s$. (The bilinear transformation in section 4 relates its analog prototype's frequency to $\omega$ differently.) The DTFT is $H(e^{j\omega })={\sum }_{n}h[n]e^{-j\omega n}$. It is $2\pi$-periodic, so $\omega =\pi$ (half the sampling rate) is the highest digital frequency. The z-transform is the bilateral one. Filter coefficients are real. Every filter here is causal, so its ROC is $|z|>r_{\max }$, where $r_{\max }$ is the largest pole radius. A causal filter with a rational $H(z)$ is BIBO stable exactly when all its poles, after cancelling common factors, lie inside $|z|=1$. An FIR filter has length $N$ (order $N-1$); in section 5 only, $N$ is the order of an analog prototype. The phase is $\theta (\omega )=\angle H(e^{j\omega })$. The group delay is ${\tau }_g(\omega )=-d\theta /d\omega$, in samples, taken wherever $H(e^{j\omega })\ne 0$. Jumps of $2\pi$ from wrapping, and jumps of $\pi$ where a real amplitude changes sign, are ignored.

A digital filter is an LTI system described by a constant-coefficient difference equation. If the output depends only on present and past inputs, the impulse response ends after a finite number of samples: a finite impulse response (FIR) filter. If the output also depends on past outputs (feedback), the impulse response in general never ends: an infinite impulse response (IIR) filter.

Why a causal, stable IIR filter cannot have exactly linear phase. The usual way to get exact linear phase is symmetry: $h[n]=\pm h[2\alpha -n]$ about some point $n=\alpha$. Combined with causality ($h[n]=0$ for $n<0$), this forces $h[n]=0$ for $n>2\alpha$, so the response is finite. More generally, exact linear phase makes a real-coefficient rational $H(z)$ satisfy $H(z)=\pm z^{-2\alpha }H(z^{-1})$. So every non-zero pole $p$ comes with a pole at $1/p$, and a pole inside the unit circle has a partner outside it. A causal filter with a pole outside the unit circle is unstable, so only an FIR filter (all poles at $z=0$) escapes. A causal, stable IIR filter can only approximate linear phase. Choose FIR when the waveform shape must be preserved (every frequency delayed by the same number of samples). Choose IIR when a sharp magnitude response is needed with little computation.

An FIR filter has (generalised) linear phase if its coefficients are symmetric or antisymmetric about the midpoint $\alpha =(N-1)/2$. The symmetry pulls out a pure delay $e^{-j\omega \alpha }$ and leaves a real amplitude function $A(\omega )$. $A(\omega )$ may change sign. Each sign change adds a jump of $\pi$ to the phase but does not change the group delay. Two choices of symmetry times two parities of length give four types.

Where the zeros lie. From $H(z)=\pm z^{-(N-1)}H(z^{-1})$, if $z_0$ is a zero then so is $1/z_0$. Because $h[n]$ is real, $z_0^{\ast }$ and $1/z_0^{\ast }$ are zeros too. So a general complex zero comes in a group of four, $\{z_0,z_0^{\ast },1/z_0,1/z_0^{\ast }\}$. A complex zero on the unit circle comes as a conjugate pair (there $1/z_0^{\ast }=z_0$). A real zero other than $\pm 1$ comes as a pair $\{r,1/r\}$. Zeros at $z=\pm 1$ can occur singly. So a linear-phase FIR filter with a zero strictly inside the unit circle always has its mirror image outside it.

The ideal lowpass response (gain 1 for $|\omega |<{\omega }_c$, 0 for ${\omega }_c<|\omega |\le \pi$) has an infinitely long, non-causal impulse response. The window method delays it by $\alpha =(N-1)/2$ so that it is centred on the window. It then multiplies it by a length-$N$ window $w[n]$ that is zero outside $0\le n\le N-1$. The result is causal, finite and linear phase (Type I or II), because both the delayed ideal response and the window are symmetric about $\alpha$.

Reading the table (all values approximate, for large $N$). Going down the table, the side lobes fall, giving better stopband attenuation. But the main lobe widens, giving a wider transition band for the same $N$. The transition width is roughly proportional to $1/N$, so increasing $N$ sharpens the edge. The attenuation, however, is fixed by the shape of the window and does not improve with $N$. So choose the window from the required stopband attenuation, then choose $N$ from the required transition width.

For instance, take ${\omega }_c=\pi /4$, $N=7$ ($\alpha =3$) and a rectangular window: $h[3]=0.25$, $h[2]=h[4]=\sin (\pi /4)/\pi =0.2251$, $h[1]=h[5]=1/(2\pi )=0.1592$, and $h[0]=h[6]=\sin (3\pi /4)/(3\pi )=0.0750$. The DC gain is $\sum h[n]=1.169$, not 1: the samples of $h_d[n]$ that the window discards sum to $-0.169$.

Between the samples the response can ripple strongly next to an abrupt jump from 1 to 0. Placing one or two intermediate-valued transition samples in the transition band trades a slightly wider transition for much lower stopband ripple. Frequency sampling is convenient when the desired response is specified at the DFT frequencies $2\pi k/N$.

IIR filters are usually designed by taking an analog prototype $H_a(s)$ (Butterworth, Chebyshev, elliptic) and mapping the $s$-plane to the $z$-plane. A good mapping sends the $j\Omega$ axis onto the unit circle, so the frequency response carries over. It also sends the left half-plane inside the unit circle, so stability carries over.

A stable analog pole ($\mathrm{Re}p<0$) gives $|e^{pT}|=e^{\mathrm{Re}(p)T}<1$, so impulse invariance preserves stability. But the map $z=e^{sT}$ is many-to-one: every horizontal strip of height $2\pi /T$ in the $s$-plane lands on the whole $z$-plane except the origin. Highpass and bandstop prototypes do not fall to zero at high frequency, so their shifted copies overlap badly. Impulse invariance is therefore unsuitable for highpass and bandstop filters. It is used for band-limited lowpass and bandpass designs, or when the sampled shape of the impulse response itself must be kept.

Comparing prototypes for the same specification. Of these, Butterworth needs the highest order. Chebyshev Type I (ripple in the passband) and Type II (monotonic passband, ripple in the stopband) need a lower order, the same for both. Elliptic (ripple in both bands) needs the lowest. Butterworth has the gentlest phase variation near the band edge; Chebyshev and elliptic buy sharpness with ripple and more phase nonlinearity. Both mapping methods keep the order: an $N$th-order prototype gives an $N$th-order digital filter.

On the unit circle $z=e^{j\omega }$, the factor $|e^{j\omega }-z_k|$ is the distance from the point at angle $\omega$ to a zero, and $|e^{j\omega }-p_k|$ the distance to a pole. The gain at $\omega$ is the product of the zero distances divided by the product of the pole distances, times the constant gain. So a zero on the unit circle at angle ${\omega }_0$ gives a null at ${\omega }_0$, and a pole close to the unit circle at angle ${\omega }_0$ gives a peak there. The point $z=1$ is $\omega =0$ (DC) and $z=-1$ is $\omega =\pi$.

A realisation is a block diagram built from adders, multipliers and unit delays ($z^{-1}$). Different structures realise the same $H(z)$ but differ in memory (delays), arithmetic (multipliers) and sensitivity to coefficient rounding. Below, an IIR filter is $H(z)=B(z)/A(z)$ with numerator order $M$ and denominator order $K$ (as in section 1), and an FIR filter has length $N$.

Counting multipliers. The counts above treat every $b_k$ and $a_k$ as a general number. A coefficient equal to $0$ needs no branch, and one equal to $\pm 1$ needs no multiplier (only an adder or subtractor). So for a specific $H(z)$, count only the coefficients that are not $0$ or $\pm 1$. The number of delays in a canonic structure equals the order of $H(z)$, which is $\max (M,K)$ after common factors have been cancelled.