Difference Equations and Digital Filter System : MALMIJAL-M Blog

Signal Processing Concepts and Engineering Insights.

Explore signal processing concepts, algorithm comparisons, and practical engineering insights.
Topics include FFT vs STFT, FRF analysis, filtering techniques, and other signal processing methods used in real engineering workflows.

Blog

Difference Equations and Digital Filter System

In digital signal processing, filters are not described by differential equations (like in analog systems), but by Difference Equations.

These equations define how each output sample depends on

Current input
Past inputs
Past outputs

Difference Equations and Digital Filter System

Mathematical Expression

Where,

x[n]: input signal
y[n]: output signal
b_r: feedforward coefficients
a_k: feedback coefficients

FIR vs IIR

FIR (Finite Impulse Response)

No feedback
Always stable
Possible to linear phase
FIR filter coefficients are the impulse response
e.g., M-point MA (Moving Average) → Low pass FIR filter

IIR (Infinite Impulse Response)

Feedback
Efficient (requires fewer coefficients)
Stability must be considered
e.g., EMA (Exponential Moving Average) → 1st order Low pass IIR filter (α: smoothing factor)

FRF of MA(FIR) and EMA(IIR)

Impulse responses of MA (FIR) and EMA (IIR: exponential decay)

FRF of MA(FIR) and EMA(IIR)

FRF of MA (FIR: linear phase) and EMA (IIR)

Key Insight

Difference Equation = Discrete System Model
Analog system: Differential Equation

Digital system: Difference Equation

Signal Processing Concepts and Engineering Insights.

Category

Systems, Filters & ModelingDifference Equations and Digital Filter System

Difference Equations and Digital Filter System

Mathematical Expression

FIR vs IIR

FIR (Finite Impulse Response)

IIR (Infinite Impulse Response)

Key Insight

Difference Equation = Discrete System Model

Suggested Further Reading