7.6 Filters

7.6 Filters

1. Filter Applications and Ideal Filter Characteristics

1.1 Filter Applications

1. Signal Processing:

   • Noise reduction and signal enhancement.

   • Signal separation (demultiplexing).

   • Anti-aliasing before sampling.

2. Communications:

   • Channel selection in radios.

   • Modulation and demodulation.

   • Equalization to compensate channel distortion.

3. Audio Processing:

   • Graphic equalizers.

   • Crossover networks in speakers.

   • Noise cancellation.

4. Image Processing:

   • Edge detection.

   • Image smoothing/enhancement.

   • Pattern recognition.

5. Control Systems:

   • Noise filtering in sensor measurements.

   • Loop shaping for stability.

6. Biomedical:

   • ECG/EEG signal conditioning.

   • Removing power line interference.

1.2 Ideal Filter Characteristics

1. Ideal Low-Pass Filter:

   • $H(j\omega) = \begin{cases} 1 & |\omega| < \omega_c \\ 0 & |\omega| > \omega_c \end{cases}$

   • Passband: $-\omega_c < \omega < \omega_c$

   • Stopband: $|\omega| > \omega_c$

   • Impulse response: $h(t) = \frac{\omega_c}{\pi} \text{sinc}(\omega_c t)$

2. Ideal High-Pass Filter:

   • $H(j\omega) = \begin{cases} 0 & |\omega| < \omega_c \\ 1 & |\omega| > \omega_c \end{cases}$

   • $H_{HP}(j\omega) = 1 - H_{LP}(j\omega)$

3. Ideal Band-Pass Filter:

   • $H(j\omega) = \begin{cases} 1 & \omega_1 < |\omega| < \omega_2 \\ 0 & \text{otherwise} \end{cases}$

   • Center frequency: $\omega_0 = \frac{\omega_1 + \omega_2}{2}$

   • Bandwidth: $BW = \omega_2 - \omega_1$

4. Ideal Band-Stop (Notch) Filter:

   • $H(j\omega) = \begin{cases} 0 & \omega_1 < |\omega| < \omega_2 \\ 1 & \text{otherwise} \end{cases}$

   • $H_{BS}(j\omega) = 1 - H_{BP}(j\omega)$

5. Problems with Ideal Filters:

   • Non-causal impulse response (extends to $t < 0$).

   • Infinite duration impulse response.

   • Physically unrealizable.

   • Sharp transitions cause Gibbs phenomenon.

2. Digital vs. Analog Filters

2.1 Analog Filters

1. Characteristics:

   • Process continuous-time signals.

   • Implemented with RLC components, op-amps.

   • Operate in real-time.

2. Advantages:

   • No quantization noise.

   • High speed (limited only by components).

   • No aliasing issues.

   • Can handle very high frequencies.

3. Disadvantages:

   • Component tolerances affect performance.

   • Temperature sensitivity.

   • Difficult to tune/adjust.

   • Bulky for low frequencies.

2.2 Digital Filters

1. Characteristics:

   • Process discrete-time signals.

   • Implemented in software or digital hardware.

   • Operate on sampled data.

2. Advantages:

   • High accuracy and reproducibility.

   • Easy to modify (programmable).

   • No component aging or temperature drift.

   • Can implement complex filters.

   • Linear phase achievable.

3. Disadvantages:

   • Limited by sampling rate.

   • Quantization errors.

   • Finite word-length effects.

   • Computational delay.

   • Aliasing if not properly designed.

2.3 Comparison Summary

Aspect	Analog Filters	Digital Filters
Signal Type	Continuous-time	Discrete-time
Implementation	Hardware (RLC, op-amps)	Software/hardware
Accuracy	Limited by components	Limited by word length
Flexibility	Fixed by design	Programmable
Frequency Range	High (GHz)	Limited by sampling
Cost	Increases with precision	Decreases with technology
Stability	Component dependent	Algorithm dependent

3. Active vs. Passive Filters

3.1 Passive Filters

1. Components: Resistors (R), Capacitors (C), Inductors (L).

2. No External Power: Use only passive components.

3. Common Types:

   • RC filters: First-order, no inductors.

   • RL filters: First-order.

   • RLC filters: Second-order, can resonate.

   • LC filters: Second-order, no resistors.

4. Frequency Response Characteristics:

   • Low-pass RC: $H(j\omega) = \frac{1}{1 + j\omega RC}$

   • High-pass RC: $H(j\omega) = \frac{j\omega RC}{1 + j\omega RC}$

   • Cutoff frequency: $\omega_c = \frac{1}{RC}$

   • Roll-off: 20 dB/decade per pole.

5. Advantages:

   • Simple and reliable.

   • No power supply needed.

   • Good for high frequencies.

   • No noise from active components.

6. Disadvantages:

   • Signal attenuation (loss).

   • Impedance matching issues.

   • Inductors large at low frequencies.

   • Loading effects.

3.2 Active Filters

1. Components: Op-amps with resistors and capacitors (no inductors).

2. External Power: Requires power supply for op-amps.

3. Common Topologies:

   • Sallen-Key: Popular second-order configuration.

   • Multiple Feedback (MFB).

   • State Variable: Can implement all filter types.

   • Biquad: Second-order section.

4. Frequency Response Characteristics:

   • Can have gain (>1).

   • High input impedance, low output impedance.

   • No loading effects between stages.

   • Precise control of parameters.

5. Advantages:

   • Gain available (amplification).

   • No inductors (compact size).

   • Good isolation between stages.

   • Tunable (via variable resistors).

   • Can implement complex filters.

6. Disadvantages:

   • Limited bandwidth (op-amp limitations).

   • Requires power supply.

   • Noise from active components.

   • Stability concerns.

   • More complex than passive.

3.3 Comparison Summary

Aspect	Passive Filters	Active Filters
Power	No external power	Requires power
Components	R, L, C	R, C, Op-amps
Gain	Attenuation only	Gain possible
Size	Large (inductors)	Compact
Frequency	Good for high freq	Limited by op-amps
Cost	Low for simple	Higher
Noise	Thermal only	Op-amp noise

4. Analog Filter Approximations

4.1 Butterworth Filter (Maximally Flat)

1. Magnitude Response: $|H(j\omega)|^2 = \frac{1}{1 + (\omega/\omega_c)^{2N}}$ where $N$ = filter order, $\omega_c$ = cutoff frequency.

2. Characteristics:

   • Maximally flat in passband (no ripple).

   • Monotonic in both passband and stopband.

   • Roll-off: 20N dB/decade.

   • Group delay not constant.

3. Pole Locations:

   • Equally spaced on circle of radius $\omega_c$.

   • In left half-plane (for stability).

   • Angles: $\theta_k = \frac{\pi}{2} + \frac{(2k-1)\pi}{2N}$, $k=1,2,...,N$

4. Transfer Function: $H(s) = \frac{K\omega_c^N}{\prod_{k=1}^{N} (s - s_k)}$ where $s_k$ are the poles.

5. Design Steps:

   1. Determine $N$ from specifications.

   2. Find pole locations.

   3. Form transfer function.

   4. Realize with circuit.

4.2 Chebyshev Filters

4.2.1 Type I (Equiripple in Passband)

1. Magnitude Response: $|H(j\omega)|^2 = \frac{1}{1 + \epsilon^2 T_N^2(\omega/\omega_c)}$ where $T_N(x)$ = Chebyshev polynomial of order N.

2. Characteristics:

   • Equiripple in passband.

   • Monotonic in stopband.

   • Sharper roll-off than Butterworth for same order.

   • Ripple controlled by $\epsilon$.

3. Parameters:

   • Passband ripple ($\delta$ in dB): $\epsilon = \sqrt{10^{\delta/10} - 1}$

   • Order $N$ determined by stopband requirements.

4.2.2 Type II (Equiripple in Stopband)

1. Magnitude Response: $|H(j\omega)|^2 = \frac{1}{1 + \epsilon^2 \left[\frac{T_N(\omega_s/\omega_c)}{T_N(\omega_s/\omega)}\right]^2}$

2. Characteristics:

   • Monotonic in passband.

   • Equiripple in stopband.

   • Better phase response than Type I.

4.3 Comparison of Approximations

Filter Type	Passband	Stopband	Roll-off	Phase Response
Butterworth	Flat	Monotonic	Moderate	Nonlinear
Chebyshev I	Ripple	Monotonic	Steep	Nonlinear
Chebyshev II	Monotonic	Ripple	Steep	Better than I
Elliptic	Ripple	Ripple	Steepest	Worst
Bessel	Flat	Poor	Gentle	Linear

4.4 Filter Order Determination

For Butterworth: $N \geq \frac{\log\left[\frac{10^{A_s/10} - 1}{10^{A_p/10} - 1}\right]}{2\log(\omega_s/\omega_p)}$ where:

• $A_p$ = passband attenuation (dB)

• $A_s$ = stopband attenuation (dB)

• $\omega_p$ = passband edge

• $\omega_s$ = stopband edge

5. Digital Filter Fundamentals

5.1 Digital Filter Representation

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

• $x[n]$ = input

• $y[n]$ = output

• $b_k$ = feedforward coefficients

• $a_k$ = feedback coefficients

5.2 Transfer Function

$H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}}$

5.3 Frequency Response

$H(e^{j\Omega}) = H(z)\big|_{z=e^{j\Omega}}$

• Periodic with period $2\pi$.

• $\Omega = \omega T$ = normalized frequency.

6. Finite Impulse Response (FIR) Filters

6.1 Definition and Structure

1. Difference Equation: $y[n] = \sum_{k=0}^{M} b_k x[n-k]$

   • No feedback terms ($a_k = 0$ for $k \geq 1$).

2. Impulse Response:

   • Finite length: $h[n] = b_n$ for $n = 0,1,...,M$.

   • Length = $M+1$ (order = $M$).

3. Transfer Function: $H(z) = \sum_{k=0}^{M} h[k] z^{-k}$

   • Only zeros (except possibly at origin).

   • Always stable (no poles except at z=0).

6.2 FIR Filter Design Methods

1. Window Method:

   • Truncate ideal impulse response with window.

   • $h[n] = h_d[n] \cdot w[n]$ for $n = -M/2,...,M/2$.

   • Common windows: Rectangular, Hamming, Hanning, Blackman.

2. Frequency Sampling Method:

   • Specify desired frequency response at equidistant points.

   • Compute inverse DFT for filter coefficients.

3. Optimal (Parks-McClellan) Method:

   • Minimax design using Remez exchange algorithm.

   • Optimal in Chebyshev sense.

   • Produces equiripple error.

6.3 Linear Phase FIR Filters

For linear phase: $h[n] = \pm h[M-n]$ Four types:

1. Type I: Symmetric, odd length (M even)

   • Can implement all filter types.

2. Type II: Symmetric, even length (M odd)

   • H(π) = 0, cannot implement high-pass.

3. Type III: Anti-symmetric, odd length

   • H(0) = H(π) = 0, suitable for Hilbert transformers.

4. Type IV: Anti-symmetric, even length

   • H(0) = 0, suitable for differentiators.

6.4 Advantages of FIR Filters

1. Always stable.

2. Linear phase achievable.

3. Simple implementation.

4. No limit cycles.

5. Good numerical properties.

6.5 Disadvantages of FIR Filters

1. High order needed for sharp cutoff.

2. Large delay (group delay = M/2 samples).

3. Computationally intensive.

7. Infinite Impulse Response (IIR) Filters

7.1 Definition and Structure

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

   • Contains feedback terms.

2. Transfer Function: $H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}}$

   • Both poles and zeros.

3. Canonical Forms:

   • Direct Form I: Separate feedforward and feedback.

   • Direct Form II: Minimal storage.

   • Cascade Form: Series of second-order sections.

   • Parallel Form: Parallel second-order sections.

7.2 IIR Filter Design Methods

1. Analog-to-Digital Transformation:

   • Impulse Invariance: Sample analog impulse response.

     • Preserves time response.

     • May cause aliasing.

   • Bilinear Transform: $s = \frac{2}{T} \frac{1-z^{-1}}{1+z^{-1}}$

     • No aliasing.

     • Warps frequency axis.

2. Direct Digital Design:

   • Least squares method.

   • Prony's method.

7.3 Stability Considerations

1. Stability Condition: All poles inside unit circle (|poles| < 1).

2. Stability Test:

   • Compute poles of H(z).

   • Jury's stability test.

   • Schur-Cohn test.

7.4 Advantages of IIR Filters

1. Lower order for same specifications.

2. Less computation per output sample.

3. Can match analog filters.

4. Efficient implementation.

7.5 Disadvantages of IIR Filters

1. Nonlinear phase (except all-pass).

2. Potential instability.

3. Limit cycles due to quantization.

4. Sensitivity to coefficient quantization.

8. FIR vs IIR Comparison

Aspect	FIR Filters	IIR Filters
Impulse Response	Finite duration	Infinite duration
Stability	Always stable	Can be unstable
Phase	Linear phase possible	Nonlinear phase
Order	High for sharp cutoff	Low for sharp cutoff
Computation	More per output	Less per output
Delay	Large	Small
Design Methods	Window, optimal	Bilinear, impulse invariance
Implementation	Simple	More complex
Applications	Where phase important	Where efficiency important

9. Important Formulas

9.1 Analog Filter Design

1. Butterworth: $|H(j\omega)|^2 = \frac{1}{1 + (\omega/\omega_c)^{2N}}$

2. Chebyshev Type I: $|H(j\omega)|^2 = \frac{1}{1 + \epsilon^2 T_N^2(\omega/\omega_c)}$

3. Order Calculation: $N \geq \frac{\log[(10^{A_s/10}-1)/(10^{A_p/10}-1)]}{2\log(\omega_s/\omega_p)}$

9.2 Digital Filter Structures

1. FIR Direct Form: $y[n] = \sum_{k=0}^{M} h[k] x[n-k]$

2. IIR Direct Form II: $w[n] = x[n] - \sum_{k=1}^{N} a_k w[n-k]$ $y[n] = \sum_{k=0}^{M} b_k w[n-k]$

9.3 Transformations

1. Bilinear Transform: $s = \frac{2}{T} \frac{1-z^{-1}}{1+z^{-1}}$ Prewarping: $\omega_a = \frac{2}{T} \tan\left(\frac{\omega_d T}{2}\right)$

2. Impulse Invariance: $h[n] = T h_c(nT)$

9.4 Filter Specifications

1. Passband ripple: $\delta_p$ (linear) or $A_p$ (dB).

2. Stopband attenuation: $\delta_s$ (linear) or $A_s$ (dB).

3. Cutoff frequency: $\omega_c$ (3-dB point).

4. Transition band: $\omega_s - \omega_p$.

9.5 Window Functions (Length M+1)

1. Rectangular: $w[n] = 1$

2. Hamming: $w[n] = 0.54 - 0.46\cos(2\pi n/M)$

3. Hanning: $w[n] = 0.5 - 0.5\cos(2\pi n/M)$

4. Blackman: $w[n] = 0.42 - 0.5\cos(2\pi n/M) + 0.08\cos(4\pi n/M)$

9.6 Linear Phase Conditions

For FIR filter of length L = M+1:

1. Type I (M even): $h[n] = h[M-n]$

2. Type II (M odd): $h[n] = h[M-n]$

3. Type III (M even): $h[n] = -h[M-n]$

4. Type IV (M odd): $h[n] = -h[M-n]$

9.7 Group Delay

1. FIR with linear phase: $\tau_g = M/2$ samples.

2. General: $\tau_g(\Omega) = -\frac{d}{d\Omega} \angle H(e^{j\Omega})$