Digital Filter – Introduction

What is a system?

A system is a rule that maps an input signal to an output signal. In continuous time and discrete time we depict and denote: \[ x(t)\ \longrightarrow\ y(t),\qquad x[n]\ \longrightarrow\ y[n]. \] These are the standard input–output representations used throughout the text.

Input–output operator notation

We often write the system as an operator ( {} ): \[ y(t)=\mathcal{T}\{x(t)\},\qquad y[n]=\mathcal{T}\{x[n]\}. \] Block diagrams are used to represent systems and interconnections (series/cascade and parallel).

Overview: what is an input–output relation?

A system specifies how an input signal produces an output signal: \[ x(t)\ \longrightarrow\ y(t),\qquad x[n]\ \longrightarrow\ y[n]. \] We study types of relations used in analysis and design: - Memoryless (static) mappings - Differential/difference-equation descriptions - Convolution (LTI) - Transform-domain forms (frequency, Laplace, (z)) - State–space (first-order vector form) - Time-varying vs time-invariant, linear vs nonlinear

Memoryless (static) relations

A memoryless or static system relates input and output at the same instant: \[ y(t)=F\!\big(x(t)\big),\qquad y[n]=F\!\big(x[n]\big). \] Examples: (y=|x|), (y=x^2), saturation and clipping nonlinearities. These are common as pointwise nonlinear stages preceding or following LTI blocks.

Linear constant-coefficient differential equations (CT)

Many continuous-time LTI systems are described by an LCCDE: \[ \sum_{k=0}^{N} a_k\,\frac{d^{\,k} y(t)}{dt^{\,k}} \;=\; \sum_{m=0}^{M} b_m\,\frac{d^{\,m} x(t)}{dt^{\,m}}, \qquad a_0\neq 0. \] Coefficients (a_k,b_m) are constants for time invariance. This form covers standard electrical/mechanical systems and filters.

Linear constant-coefficient difference equations (DT)

Discrete-time LTI systems admit an LCCD equation: \[ \sum_{k=0}^{N} a_k\,y[n-k]\;=\;\sum_{m=0}^{M} b_m\,x[n-m],\qquad a_0\neq 0. \] This representation includes IIR/FIR digital filters and many algorithmic recursions.

Convolution relations (LTI)

For LTI systems the input–output relation is convolution with the impulse response: \[ y(t)=\int_{-\infty}^{\infty} h(\tau)\,x(t-\tau)\,d\tau,\qquad y[n]=\sum_{k=-\infty}^{\infty} h[k]\;x[n-k]. \] Here (h(t)) or (h[n]) is the output to a unit impulse. Causality for LTI corresponds to (h(t)=0) for (t<0) (or (h[n]=0) for (n<0)).

Frequency-domain input–output (Fourier)

When Fourier transforms exist, convolution becomes multiplication: \[ Y(\omega)=H(\omega)\,X(\omega),\qquad Y\!\big(e^{j\omega}\big)=H\!\big(e^{j\omega}\big)\,X\!\big(e^{j\omega}\big). \] (H) is the frequency response (CTFT/DTFT). Magnitude (|H|) scales, phase (H) shifts/warps timing.

Laplace and (z)-transform forms

With Laplace and (z)-transforms (for appropriate regions of convergence): \[ Y(s)=H(s)\,X(s),\qquad Y(z)=H(z)\,X(z), \] and for LCCDE/LCCD systems \[ H(s)=\frac{b_0+b_1 s+\cdots+b_M s^M}{a_0+a_1 s+\cdots+a_N s^N},\qquad H(z)=\frac{b_0+b_1 z^{-1}+\cdots+b_M z^{-M}}{a_0+a_1 z^{-1}+\cdots+a_N z^{-N}}. \] These rational forms support pole–zero analysis and stability checks.

State–space (first-order vector) description

An equivalent input–output formulation uses state variables: \[ \dot{\mathbf{s}}(t)=\mathbf{A}\,\mathbf{s}(t)+\mathbf{b}\,x(t),\qquad y(t)=\mathbf{c}^\top\mathbf{s}(t)+d\,x(t). \] Discrete time: \[ \mathbf{s}[n+1]=\mathbf{A}\,\mathbf{s}[n]+\mathbf{b}\,x[n],\qquad y[n]=\mathbf{c}^\top\mathbf{s}[n]+d\,x[n]. \] This first-order form is algebraically equivalent to LCCDE/LCCD for LTI systems.

Time-varying vs time-invariant relations

• Time-invariant (TI): coefficients and operators do not change with time/index; shifts commute with the system.
• Time-varying (TV): coefficients depend on (t) or (n): \[ \sum_{k=0}^{N} a_k(t)\,\frac{d^{\,k} y(t)}{dt^{\,k}} =\sum_{m=0}^{M} b_m(t)\,\frac{d^{\,m} x(t)}{dt^{\,m}}. \] TV models arise in modulated systems and adaptive filtering.

Linear vs nonlinear relations

• Linear: superposition holds — additivity and homogeneity.
• Nonlinear: violates superposition; examples include squares, rectifiers, saturators, quantizers. Nonlinear stages are often analyzed locally or via small-signal linearization around an operating point.

Stochastic input–output (LTI summary)

For wide-sense stationary (WSS) inputs to an LTI system: \[ \mu_y=\mu_x\,H(0)\ \text{(when defined)},\qquad S_{yy}(\omega)=\big|H(\omega)\big|^2\,S_{xx}(\omega). \] This links input and output statistics through (H), supporting noise and SNR analysis.

Interconnections (examples)

• Series (cascade): output of System 1 feeds System 2.
• Parallel: both systems process the same input; outputs are summed. These patterns are foundational for building complex systems from simpler ones.

Property: Memoryless vs. with memory

• Memoryless: the output at an instant depends only on the input at that same instant.
• With memory: the output depends on past/future values (e.g., accumulators, averagers). Example: the summer/accumulator (running sum) and its inverse (first difference) illustrate systems with memory: \[ y[n]=\sum_{k=-\infty}^{n} x[k],\qquad \text{inverse: } y[n]=x[n]-x[n-1]. \] Both are causal (see next slide)

Property: Causality (general and LTI)

Causal: output depends only on present/past input values. For LTI systems, causality is characterized by the impulse response: \[ \text{Discrete time: } h[n]=0\ \text{for } n<0;\qquad \text{Continuous time: } h(t)=0\ \text{for } t<0. \] Under these conditions, the convolution reduces to depend only on past/present input.

Property: Stability (BIBO)

Bounded-Input Bounded-Output (BIBO) stability: bounded input implies bounded output. For LTI systems: \[ \sum_{k=-\infty}^{\infty} |h[k]| < \infty \quad \Longleftrightarrow \quad \text{discrete-time LTI is stable}, \] \[ \int_{-\infty}^{\infty} |h(t)|\,dt < \infty \quad \Longleftrightarrow \quad \text{continuous-time LTI is stable}. \] These are necessary and sufficient conditions.

Property: Linearity (superposition)

A system is linear if it satisfies additivity and homogeneity: for any signals (x_1,x_2) and scalar (c), \[ \mathcal{T}\{x_1+x_2\}=\mathcal{T}\{x_1\}+\mathcal{T}\{x_2\},\qquad \mathcal{T}\{c\,x\}=c\,\mathcal{T}\{x\}. \] (These two conditions together are equivalent to linearity.)

Property: Time invariance (definition)

A system is time-invariant if a shift in the input produces an identical shift in the output: \[ \mathcal{T}\{x(t-t_0)\}=y(t-t_0),\ \text{whenever}\ y(t)=\mathcal{T}\{x(t)\}. \] Analogously in discrete time with shifts by integer indices. (Definition used throughout the text in system properties and LTI analysis.)

Property: Invertibility

A system is invertible if distinct inputs produce distinct outputs; equivalently, there exists an inverse system that recovers the input from the output. Example pair (discrete time): the accumulator and the first-difference operator are inverses: \[ y[n]=\sum_{k=-\infty}^{n} x[k] \quad \Longleftrightarrow \quad x[n]=y[n]-y[n-1]. \]

Additional note: Interconnections and LTI analysis

For LTI systems, interconnections admit simple algebraic descriptions via transforms: e.g., in the Laplace domain, series and parallel lead to product and sum of system functions, respectively: \[ H_{\text{series}}(s)=H_1(s)H_2(s),\qquad H_{\text{parallel}}(s)=H_1(s)+H_2(s). \]

Digital Filter – Introduction

Digital Filter – Introduction

Convolution

Example: RC Low-Pass Filter Response

Consider a first-order RC circuit with impulse response \(h(t) = \frac{1}{RC} e^{-t/RC} u(t)\). If the input is a rectangular pulse \(x(t) = u(t) - u(t-T)\), the output \(y(t)\) is:

\[y(t) = \int_{0}^{t} x(\tau) \frac{1}{RC} e^{-(t-\tau)/RC} d\tau\]

For \(0 \le t \le T\): \[y(t) = 1 - e^{-t/RC}\]

For \(t > T\): \[y(t) = (1 - e^{-T/RC}) e^{-(t-T)/RC}\]

RC Filter: Impulse Response Derivation

The voltage across a capacitor in an RC circuit is governed by the first-order differential equation: \[RC \frac{dy(t)}{dt} + y(t) = x(t)\]

Applying the Laplace Transform with zero initial conditions: \[Y(s)(RCs + 1) = X(s) \implies H(s) = \frac{Y(s)}{X(s)} = \frac{1/RC}{s + 1/RC}\]

The impulse response \(h(t)\) is the inverse Laplace Transform \(\mathcal{L}^{-1}\{H(s)\}\): \[h(t) = \frac{1}{RC} e^{-\frac{t}{RC}} u(t)\]

Time instants

Analytical Evaluation: Interval \(0 \le t \le T\)

For a rectangular pulse \(x(t) = 1\) for \(0 \le t \le T\), and \(h(t) = \alpha e^{-\alpha t} u(t)\) where \(\alpha = 1/RC\):

\[y(t) = \int_{0}^{t} (1) \cdot \alpha e^{-\alpha(t-\tau)} d\tau\]

Factoring out terms independent of \(\tau\): \[y(t) = \alpha e^{-\alpha t} \int_{0}^{t} e^{\alpha \tau} d\tau = \alpha e^{-\alpha t} \left[ \frac{1}{\alpha} e^{\alpha \tau} \right]_{0}^{t}\]

Evaluating the limits: \[y(t) = e^{-\alpha t} (e^{\alpha t} - 1) = 1 - e^{-\alpha t}\]

Analytical Evaluation: Interval \(t > T\)

When the input pulse ends (\(t > T\)), the integration limits are constrained by the pulse width \([0, T]\):

\[y(t) = \int_{0}^{T} (1) \cdot \alpha e^{-\alpha(t-\tau)} d\tau = \alpha e^{-\alpha t} \int_{0}^{T} e^{\alpha \tau} d\tau\]

Solving the integral: \[y(t) = \alpha e^{-\alpha t} \left[ \frac{1}{\alpha} e^{\alpha \tau} \right]_{0}^{T} = e^{-\alpha t} (e^{\alpha T} - 1)\]

Rearranging to show the exponential decay from the value at \(t=T\): \[y(t) = (1 - e^{-\alpha T}) e^{-\alpha(t-T)}\]

Python Implementation: Continuous-Time Emulation

Numerical approximation of the continuous convolution using high-density sampling.

import numpy as np
import matplotlib.pyplot as plt

# Parameters
RC = 0.5
alpha = 1/RC
T = 2.0
dt = 0.01
t = np.arange(0, 5, dt)

# Signals
x = np.where((t >= 0) & (t <= T), 1.0, 0.0)
h = alpha * np.exp(-alpha * t)

# Convolution (scaled by dt to approximate integral)
y = np.convolve(x, h)[:len(t)] * dt

print(f"Peak value at t=T: {y[int(T/dt)]: .4f}")

Peak value at t=T:  0.9919

print(f"Theoretical peak: {1 - np.exp(-alpha*T): .4f}")

Theoretical peak:  0.9817

Visualization: Continuous Convolution

The “Flip-Shift-Integrate” process:

Discrete-Time Convolution: Definition

In the discrete domain, the convolution sum relates the input \(x[n]\) and the impulse response \(h[n]\) of a discrete LTI system:

\[y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n - k]\]

This operation characterizes the system’s output as a weighted sum of delayed impulses. For causal signals of finite length \(L\) and \(M\), the resulting sequence \(y[n]\) has length \(L + M - 1\).

Example: Moving Average Filter

Let \(x[n] = [1, 2, 3, 2, 1]\) and a 3-point causal moving average filter \(h[n] = \frac{1}{3}[\delta[n] + \delta[n-1] + \delta[n-2]]\).

Analytical Calculation:

• \(y[0] = \frac{1}{3}(1) = 0.33\)
• \(y[1] = \frac{1}{3}(1+2) = 1.0\)
• \(y[2] = \frac{1}{3}(1+2+3) = 2.0\)
• \(y[3] = \frac{1}{3}(2+3+2) = 2.33\)
• \(y[4] = \frac{1}{3}(3+2+1) = 2.0\)

Visualization: Discrete Convolution

Mechanism of \(h[n-k]\) sliding over \(x[k]\) at \(n=2\):

Numerical Validation in Python

The following code validates the discrete convolution using numpy.convolve.

import numpy as np

# Define signals
x = np.array([1, 2, 3, 2, 1])
h = np.array([1/3, 1/3, 1/3])

# Compute convolution
y = np.convolve(x, h, mode='full')

# Display results
print(f"Input x[n]: {x}")

Input x[n]: [1 2 3 2 1]

print(f"Impulse response h[n]: {h}")

Impulse response h[n]: [0.33333333 0.33333333 0.33333333]

print(f"Output y[n]: {np.round(y, 2)}")

Output y[n]: [0.33 1.   2.   2.33 2.   1.   0.33]

# Verification of length
expected_len = len(x) + len(h) - 1
print(f"Verified Length: {len(y) == expected_len}")

Verified Length: True

Comparison of Domain Characteristics

Ejemplo 1: Convolución Continua (Planteamiento)

Considere señales analógicas definidas por tramos (no simétricas):

Señal de entrada \(x(t)\): \[x(t) = \begin{cases} t & 0 \leq t < 1 \\ 1 & 1 \leq t < 2 \\ 0 & \text{otros} \end{cases}\]

Respuesta al impulso \(h(t)\): \[h(t) = \begin{cases} 1 & 0 \leq t < 1 \\ -0.5 & 1 \leq t < 1.5 \\ 0 & \text{otros} \end{cases}\]

Objetivo: Calcular la respuesta del sistema \(y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau\).

Definición Formal de las Señales

Para un análisis riguroso, expresamos las señales asimétricas mediante funciones escalón unitario \(u(t)\):

Entrada \(x(t)\): \[x(t) = t[u(t)-u(t-1)] + 1[u(t-1)-u(t-2)]\]

Respuesta al impulso \(h(t)\): \[h(t) = 1[u(t)-u(t-1)] - 0.5[u(t-1)-u(t-1.5)]\]

El soporte de la salida \(y(t) = x(t) * h(t)\) es la suma de los soportes: \[S_y = [0+0, 2+1.5] = [0, 3.5]\]

Metodología: El Kernel Invertido

La integral de convolución es \(y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau\). El término \(h(t-\tau)\) actúa como una ventana móvil que se desplaza de izquierda a derecha según aumenta \(t\):

• Región de \(h=1\): \(\tau \in [t-1, t]\)
• Región de \(h=-0.5\): \(\tau \in [t-1.5, t-1]\)

La evaluación requiere identificar cada cambio en la intersección de estos intervalos con los tramos de \(x(\tau)\) (\([0,1]\) y \([1,2]\)).

Análisis por Intervalos (I)

1. Intervalo \(0 \leq t < 1\): Solo la parte frontal de \(h\) (\(h=1\)) entra en el soporte de \(x\). \[y(t) = \int_{0}^{t} \tau \cdot (1) d\tau = \frac{t^2}{2}\]

2. Intervalo \(1 \leq t < 1.5\): La parte \(h=1\) cubre totalmente el primer tramo de \(x\) y entra en el segundo. La parte \(h=-0.5\) entra en el primer tramo. \[y(t) = \underbrace{\int_{0}^{t-1} \tau(-0.5) d\tau}_{\text{cola de } h} + \underbrace{\int_{t-1}^{1} \tau(1) d\tau}_{\text{frente } h \text{ en tramo 1}} + \underbrace{\int_{1}^{t} (1)(1) d\tau}_{\text{frente } h \text{ en tramo 2}}\]

Análisis por Intervalos (II)

3. Intervalo \(1.5 \leq t < 2\): Ambas partes de \(h\) están dentro del soporte de \(x\). \[y(t) = \int_{t-1.5}^{t-1} \tau(-0.5) d\tau + \int_{t-1}^{1} \tau(1) d\tau + \int_{1}^{2} (1)(1) d\tau\]

4. Intervalo \(2 \leq t < 2.5\): El frente de \(h\) comienza a salir del soporte de \(x\) (\(t-1 > 1\)). \[y(t) = \int_{t-1.5}^{1} \tau(-0.5) d\tau + \int_{1}^{t-1} 1(-0.5) d\tau + \int_{t-1}^{2} (1)(1) d\tau\]

Análisis por Intervalos (III)

5. Intervalo \(2.5 \leq t < 3\): Solo la cola de \(h\) (\(h=-0.5\)) permanece en el segundo tramo de \(x\). \[y(t) = \int_{t-1.5}^{2} (1)(-0.5) d\tau = -0.5(2 - (t - 1.5))\]

6. Intervalo \(t \geq 3.5\): \[y(t) = 0\]

Ejemplo 2: Convolución Discreta (Planteamiento)

Sean las secuencias asimétricas compuestas \(x[n]\) y \(h[n]\):

Entrada \(x[n]\): \[x[n] = \{1, 3, 1.5\} \text{ para } n=0, 1, 2\]

Respuesta al impulso \(h[n]\) (Filtro Derivador Asimétrico): \[h[n] = \begin{cases} 0.5 & n=0 \\ 1 & n=1 \\ -1 & n=2 \\ 0.2 & n=3 \\ 0 & \text{otros} \end{cases}\]

Determine \(y[n] = x[n] * h[n]\) mediante el método de suma directa para sistemas LTI discretos.

Ejemplo 2: Desarrollo y Resultado

Dado que \(x[n]\) tiene longitud \(N=3\) y \(h[n]\) longitud \(M=4\), \(y[n]\) tendrá \(N+M-1 = 6\) muestras significativas (\(n=0\) hasta \(n=5\)).

Cálculo de muestras: * \(y[0] = x[0]h[0] = 1(0.5) = \mathbf{0.5}\) * \(y[1] = x[1]h[0] + x[0]h[1] = 3(0.5) + 1(1) = \mathbf{2.5}\) * \(y[2] = x[2]h[0] + x[1]h[1] + x[0]h[2] = 1.5(0.5) + 3(1) + 1(-1) = \mathbf{2.75}\) * \(y[3] = x[2]h[1] + x[1]h[2] + x[0]h[3] = 1.5(1) + 3(-1) + 1(0.2) = \mathbf{-1.3}\) * \(y[4] = x[2]h[2] + x[1]h[3] = 1.5(-1) + 3(0.2) = \mathbf{-0.9}\) * \(y[5] = x[2]h[3] = 1.5(0.2) = \mathbf{0.3}\)

Resultado: \(y[n] = \{0.5, 2.5, 2.75, -1.3, -0.9, 0.3\}\)