Digital Filters

Introduction

Digital filters are the workhorses of modern signal processing. Every time your phone processes audio, every time a medical instrument cleans up an ECG signal, every time a communication receiver separates your data from noise — digital filters are doing the heavy lifting. A digital filter takes a discrete-time input sequence and produces a discrete-time output sequence according to a specific mathematical rule, selectively modifying the frequency content of the signal.

For B.Tech students, understanding digital filters means bringing together difference equations, z-transforms, frequency response, and implementation — it is where the mathematical theory becomes tangible, implementable code running on real hardware.

What is a Digital Filter?

A digital filter is a discrete-time LTI system characterized by its difference equation:

$$y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k]$$

where $x[n]$ is the input, $y[n]$ is the output, $b_k$ are the feedforward coefficients, and $a_k$ are the feedback coefficients. Taking the z-transform:

$$H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}} = \frac{b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}}{1 + a_1 z^{-1} + \cdots + a_N z^{-N}}$$

The transfer function $H(z)$ completely characterizes the filter. Its poles and zeros determine the frequency response shape.

FIR vs IIR: The Two Families

FIR (Finite Impulse Response) Filters

When all feedback coefficients are zero ($a_k = 0$), the filter is FIR:

$$y[n] = \sum_{k=0}^{M} b_k x[n-k] = b_0 x[n] + b_1 x[n-1] + \cdots + b_M x[n-M]$$

The impulse response has exactly $M+1$ nonzero samples: $h[n] = b_n$ for $0 \leq n \leq M$.

Advantages of FIR:

• Always stable (no feedback, no poles except at $z = 0$)
• Can achieve exactly linear phase (constant group delay)
• No limit cycle oscillations in fixed-point implementations
• Simple to design using windowing or Parks-McClellan algorithm

Disadvantage: Requires many more coefficients (taps) than IIR for the same frequency selectivity. A sharp low-pass filter might need 100+ FIR taps vs. 5-10 IIR coefficients.

IIR (Infinite Impulse Response) Filters

When feedback coefficients are nonzero, the filter is IIR — its impulse response extends infinitely:

$$y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k]$$

Advantages of IIR:

• Much more computationally efficient (fewer coefficients for sharp transitions)
• Can directly implement classical analog filter designs (Butterworth, Chebyshev, Elliptic)

Disadvantages:

• Can be unstable if poles move outside the unit circle
• Cannot achieve linear phase (introduces phase distortion)
• Susceptible to limit cycles and overflow in fixed-point arithmetic

Frequency Response

The frequency response is obtained by evaluating $H(z)$ on the unit circle ($z = e^{j\omega}$):

$$H(e^{j\omega}) = |H(e^{j\omega})| \cdot e^{j\angle H(e^{j\omega})}$$

The magnitude response $|H(e^{j\omega})|$ tells us how much each frequency is amplified or attenuated. The phase response $\angle H(e^{j\omega})$ tells us how much each frequency is delayed.

For a filter to pass frequencies in the band $[\omega_1, \omega_2]$ and reject others:

• Passband: $|H(e^{j\omega})| \approx 1$ (frequencies passed through)
• Stopband: $|H(e^{j\omega})| \approx 0$ (frequencies blocked)
• Transition band: Gradual roll-off between passband and stopband

Filter Types by Frequency Selection

Low-Pass Filter (LPF)

Passes frequencies below cutoff $\omega_c$, rejects above. Used for: anti-aliasing, smoothing, noise removal.

High-Pass Filter (HPF)

Passes frequencies above cutoff, rejects below. Used for: DC removal, edge detection, baseline wander removal.

Band-Pass Filter (BPF)

Passes frequencies between $\omega_1$ and $\omega_2$. Used for: selecting a radio channel, isolating a frequency band.

Band-Stop (Notch) Filter

Rejects frequencies in a band. Used for: removing 50/60 Hz power line interference, eliminating specific tones.

Linear Phase and Its Importance

A filter has linear phase if $\angle H(e^{j\omega}) = -\alpha\omega$ (phase is proportional to frequency). This means all frequencies are delayed by the same amount (group delay = $\alpha$ samples), preserving the waveshape.

For FIR filters, linear phase is achieved when the coefficients are symmetric or antisymmetric:

• Type I: $h[n] = h[M-n]$, odd length, symmetric — can implement any filter type
• Type II: $h[n] = h[M-n]$, even length, symmetric — cannot be highpass
• Type III: $h[n] = -h[M-n]$, odd length, antisymmetric — differentiators, Hilbert transformers
• Type IV: $h[n] = -h[M-n]$, even length, antisymmetric — differentiators, Hilbert transformers

Design Methods

FIR Design: Window Method

1. Start with the ideal (unrealizable) impulse response $h_d[n]$ (e.g., sinc for low-pass)
2. Truncate to finite length by multiplying with a window $w[n]$
3. The choice of window controls the trade-off between main-lobe width (transition band) and sidelobe level (stopband attenuation)

Common windows: Rectangular (-13 dB), Hamming (-43 dB), Blackman (-58 dB), Kaiser (adjustable).

IIR Design: Bilinear Transform

1. Design an analog prototype filter (Butterworth, Chebyshev, Elliptic)
2. Apply the bilinear transformation: $s = \frac{2}{T_s} \cdot \frac{1-z^{-1}}{1+z^{-1}}$
3. This maps the entire analog frequency axis to one revolution around the unit circle

Frequency warping: $\omega_{analog} = \frac{2}{T_s}\tan(\omega_{digital}/2)$. Pre-warp critical frequencies before designing the analog prototype.

Implementation Structures

Direct Form I

Implements the difference equation directly. Requires $M + N + 1$ multipliers and $M + N$ delay elements.

Direct Form II (Transposed)

Minimizes memory by sharing delay elements between feedforward and feedback paths. Requires only $\max(M, N)$ delays.

Cascade (Second-Order Sections)

Factor $H(z)$ into second-order sections (biquads): $H(z) = \prod_{k} H_k(z)$. Each biquad has at most 2 poles and 2 zeros. This is the preferred implementation for IIR filters because it minimizes coefficient sensitivity and overflow issues.

Worked Example

Design a simple 3-tap averaging filter (low-pass FIR):

$$h[n] = \{1/3, 1/3, 1/3\}$$

$$H(z) = \frac{1}{3}(1 + z^{-1} + z^{-2})$$

Frequency response: $H(e^{j\omega}) = \frac{1}{3}e^{-j\omega}(1 + 2\cos\omega)$

This has zeros at $\omega = \pm 2\pi/3$ and magnitude that decreases with frequency — a simple low-pass behavior. The phase is linear ($-\omega$ after factoring out $e^{-j\omega}$).

Key Takeaways

• Digital filters implement difference equations that selectively modify frequency content
• FIR filters are always stable and can achieve linear phase; IIR filters are more efficient but may be unstable
• The transfer function $H(z)$ and its evaluation on the unit circle give the complete frequency response
• Filter types (LP, HP, BP, BS) are characterized by their passband and stopband regions
• FIR design uses windowed ideal responses; IIR design uses bilinear transform from analog prototypes
• Cascade biquad sections are the preferred IIR implementation for numerical robustness