5.01 Ideal Filters Practical Specifications and Tolerance Cu

Related Concepts: 5.02 Butterworth Filter Approximation | 5.03 Chebyshev, Elliptic & Bessel Filter Approximations | 5.04 Active Filter Realization & Sallen-Key RC Networks

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5.01 Ideal Filters, Practical Specifications & Tolerance Curves

Core Idea

Filtering is a fundamental frequency-domain operation designed to pass a desired band of frequency components without distortion while completely suppressing all other undesired frequency bands. While ideal continuous-time filters have “brick-wall” rectangular frequency profiles, they require non-causal sinc-shaped impulse responses that extend infinitely into negative time, making them physically unrealizable. Practical filter design therefore relies on specifying mathematical tolerance bands (passband ripple, stopband attenuation, and transition widths) that can be approximated by realizable rational transfer functions.

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1. What is Filtering?

In signal processing, filtering is defined as an operation that selectively shapes the spectrum of an input signal. A filter is an LTI system characterized by a frequency response $H(j\Omega )$ that acts as a spectral gate.

1.1 Major Applications of Filtering

Examiners frequently ask why filtering is critical in engineering. Key modern applications include:

1. Noise Suppression: Stripping high-frequency thermal noise from low-frequency physiological signals (e.g., ECG or EEG).
2. Demodulation & Baseband Recovery: Extracting message signals $m(t)$ from modulated carrier waves $s(t)$ in communication receivers.
3. Anti-Aliasing: Placing a strict low-pass filter before an Analog-to-Digital Converter (ADC) to band-limit the input signal to under $f_s/2$.
4. Frequency-Division Multiplexing (FDM): Demultiplexing multiple adjacent channels sharing a single medium by isolating individual carriers.
5. Structural Vibration Analysis: Isolating mechanical vibration frequencies of a bridge or high-rise building caused specifically by wind forces rather than passing vehicles.

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2. Mathematical Models of Ideal Analog Filters

An ideal analog filter possesses a frequency response with flat, unity-gain passbands and zero-gain stopbands, separated by discontinuous vertical cuts called brick-wall boundaries.

2.1 The Ideal Low-Pass Filter (LPF)

An ideal low-pass filter passes all frequency components below a specified cutoff frequency ${\Omega }_c$ and completely blocks all components above it: $H_{\text{LP}}(j\Omega )=\{\begin{array}{ll}1,& |\Omega |\le {\Omega }_c\\ 0,& |\Omega |>{\Omega }_c\end{array}$

Step-by-Step Impulse Response Derivation:

The impulse response $h_{\text{LP}}(t)$ is the inverse Fourier transform of $H_{\text{LP}}(j\Omega )$: $h_{\text{LP}}(t)={\mathcal{L}}^{-1}\{H_{\text{LP}}(j\Omega )\}=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{H}_{\text{LP}}(j\Omega )e^{j\Omega t}d\Omega$ Substitute the limits of the passband: $h_{\text{LP}}(t)=\frac{1}{2\pi }{\int }_{-{\Omega }_c}^{{\Omega }_c}(1)e^{j\Omega t}d\Omega =\frac{1}{2\pi }{\left[\frac{e^{j\Omega t}}{jt}\right]}_{-{\Omega }_c}^{{\Omega }_c}$ $h_{\text{LP}}(t)=\frac{e^{j{\Omega }_{c}t}-e^{-j{\Omega }_{c}t}}{2\pi jt}=\frac{\sin ({\Omega }_{c}t)}{\pi t}$ Using the normalized sinc function definition where $\text{sinc}(x)=\frac{\sin (\pi x)}{\pi x}$: $h_{\text{LP}}(t)=\frac{{\Omega }_c}{\pi }\text{sinc}\left(\frac{{\Omega }_{c}t}{\pi }\right)$

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2.2 The Ideal High-Pass Filter (HPF)

An ideal high-pass filter blocks all frequencies below ${\Omega }_c$ and passes all frequencies above it: $H_{\text{HP}}(j\Omega )=\{\begin{array}{ll}0,& |\Omega |\le {\Omega }_c\\ 1,& |\Omega |>{\Omega }_c\end{array}$ We can express the ideal high-pass response as a parallel subtraction of the low-pass block from an all-pass channel: $H_{\text{HP}}(j\Omega )=1-H_{\text{LP}}(j\Omega )$ Taking the inverse Fourier transform of both sides yields the impulse response: $h_{\text{HP}}(t)=\delta (t)-\frac{{\Omega }_c}{\pi }\text{sinc}\left(\frac{{\Omega }_{c}t}{\pi }\right)$

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2.3 The Ideal Band-Pass Filter (BPF)

An ideal band-pass filter passes frequencies situated within a restricted band between cutoff frequencies ${\Omega }_{c1}$ and ${\Omega }_{c2}$ (where ${\Omega }_{c1}<{\Omega }_{c2}$): $H_{\text{BP}}(j\Omega )=\{\begin{array}{ll}1,& {\Omega }_{c1}\le |\Omega |\le {\Omega }_{c2}\\ 0,& \text{otherwise}\end{array}$ Its impulse response is derived by taking individual passband integrals: $h_{\text{BP}}(t)=\frac{1}{2\pi }{\int }_{-{\Omega }_{c2}}^{-{\Omega }_{c1}}{e}^{j\Omega t}d\Omega +\frac{1}{2\pi }{\int }_{{\Omega }_{c1}}^{{\Omega }_{c2}}{e}^{j\Omega t}d\Omega$ $h_{\text{BP}}(t)=\frac{\sin ({\Omega }_{c2}t)-\sin ({\Omega }_{c1}t)}{\pi t}=\frac{{\Omega }_{c2}}{\pi }\text{sinc}\left(\frac{{\Omega }_{c2}t}{\pi }\right)-\frac{{\Omega }_{c1}}{\pi }\text{sinc}\left(\frac{{\Omega }_{c1}t}{\pi }\right)$

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2.4 The Ideal Band-Stop Filter (BSF)

An ideal band-stop (or band-rejection) filter blocks all frequency components lying within the band from ${\Omega }_{c1}$ to ${\Omega }_{c2}$ and passes all other frequencies: $H_{\text{BS}}(j\Omega )=\{\begin{array}{ll}1,& |\Omega |\le {\Omega }_{c1}\text{ and }|\Omega |\ge {\Omega }_{c2}\\ 0,& {\Omega }_{c1}<|\Omega |<{\Omega }_{c2}\end{array}$ Representing the band-stop filter as a parallel sum of an LPF and an HPF: $H_{\text{BS}}(j\Omega )=H_{\text{LP}}(j\Omega )+H_{\text{HP}}(j\Omega )$ Taking the inverse Fourier transform yields the impulse response: $h_{\text{BS}}(t)=\delta (t)+\frac{{\Omega }_{c1}}{\pi }\text{sinc}\left(\frac{{\Omega }_{c1}t}{\pi }\right)-\frac{{\Omega }_{c2}}{\pi }\text{sinc}\left(\frac{{\Omega }_{c2}t}{\pi }\right)$

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3. The Non-Causality & Unrealizability Proof

A critical theoretical question in ECE 2107 examinations is: “Prove that ideal filters are physically unrealizable.”

Proof of Non-Causality:

1. Definition of Causality: For an LTI system to be physically causal, its impulse response must satisfy: $h(t)=0\text{for }t<0$
2. The Ideal LPF Impulse Response: As derived in Section 2.1, the impulse response of an ideal LPF is: $h_{\text{LP}}(t)=\frac{{\Omega }_c}{\pi }\text{sinc}\left(\frac{{\Omega }_{c}t}{\pi }\right)=\frac{\sin ({\Omega }_{c}t)}{\pi t}$
3. Evaluating for Negative Time ($t<0$): For any $t<0$ (excluding the discrete zero-crossings where ${\Omega }_{c}t=-k\pi$): $h_{\text{LP}}(t)\ne 0$ {Because the Sinc function envelope decays as $1/t$, it has non-zero oscillatory tails extending all the way back to $t=-\infty$}.
4. Physical Meaning: This implies that the system produces an output response before the input impulse is applied at $t=0$. This requires anticipation of future inputs, which is physically impossible.
5. Conclusion: Therefore, the ideal filter is non-causal and cannot be physically built using real-world resistors, capacitors, and active components.

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4. Practical Analog Filter Specifications & Tolerance Curves

Since ideal “brick-wall” filters are physically impossible, practical filters must tolerate some deviation from unity gain in the passband and some non-zero leakage in the stopband. The transition from the passband to the stopband cannot be instantaneous; it must occur over a finite transition band.

4.1 Parameter Definitions

To design a realizable rational filter, we define six mandatory parameters:

• ${\Omega }_p$ (Passband Edge Frequency): The frequency bounding the end of the passband.
• ${\Omega }_s$ (Stopband Edge Frequency): The frequency defining the start of the stopband.
• ${\delta }_p$ (Passband Peak Ripple): The maximum allowed amplitude deviation from unity gain in the passband.
• ${\delta }_s$ (Stopband Peak Leakage): The maximum allowed amplitude gain in the stopband.
• ${\alpha }_p$ (Peak Passband Ripple in dB or Passband Loss): ${\alpha }_p=-20{\log }_{10}(1-{\delta }_p)\text{ dB}$
• ${\alpha }_s$ (Minimum Stopband Attenuation in dB or Stopband Loss): ${\alpha }_s=-20{\log }_{10}({\delta }_s)\text{ dB}$

4.2 Practical Filter Specifications Tolerance Graph

The following graph maps the tolerance boundaries that any practical low-pass filter’s magnitude response $|H(j\Omega )|$ must respect:

    Amplitude |H(jΩ)|
       ^
  1+δp |-------------+  \
    1  |   ~ ~ ~ ~   |   \  Passband Ripple Band
  1-δp |-------------+----\-----------------------------
       |             |     \
       |             |      \  Transition Band
       |             |       \
       |             |        \
  δs   |             |         +------------------------  <-- Stopband Floor
    0  +-------------+---------+------------------------>
       0            Ωp        Ωs                        Ω (Frequency)

Placeholder Source: Figure 5.8 (Specifications of a low-pass analog filter) — (k.Deergha Rao) signals and systems.pdf (Ch 5).

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5. Summary of Ideal Filter Responses

Filter Type	Frequency Domain Profile $H(j\Omega )$	Impulse Response $h(t)$
Low-Pass (LPF)	$1 \text{ for }	\Omega
High-Pass (HPF)	$1 \text{ for }	\Omega
Band-Pass (BPF)	$1 \text{ for } \Omega_{c1} \le	\Omega
Band-Stop (BSF)	$0 \text{ for } \Omega_{c1} <	\Omega

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6. Common Mistakes That Cost Marks

The Sinc Definition Discrepancy

Standard mathematical software (like MATLAB) utilizes the normalized sinc function: $\text{sinc}(x)=\frac{\sin (\pi x)}{\pi x}$. However, many engineering textbooks define it as the unnormalized function: $\text{Sa}(x)=\frac{\sin (x)}{x}$. In examinations, always state your definition clearly before writing out step-responses. If utilizing the normalized sinc function, the LPF impulse response must include the scaling factors: $h(t)=\frac{{\Omega }_c}{\pi }\text{sinc}\left(\frac{{\Omega }_{c}t}{\pi }\right)$.

The Inverse Log Formula Sign Trap

When solving for the peak ripple parameters ${\delta }_p$ and ${\delta }_s$ from dB specifications, remember the signs: ${\delta }_p=1-10^{-{\alpha }_p/20}\text{and}{\delta }_s=10^{-{\alpha }_s/20}$ Writing positive exponents (e.g., $10^{{\alpha }_s/20}$) results in astronomical values that make no physical sense for fractional gain limits.

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7. PYQ Bank — Verbatim Questions & Answer Plans

7.1 PYQ 2024 Question 4a [08 Marks]

Question: Draw the specifications for analog low pass filter with tolerance curve.

• Answer Plan:
  1. Sketch the exact specifications curve shown in Section 4.2, clearly labeling the vertical axis ($1+{\delta }_p$, $1-{\delta }_p$, and ${\delta }_s$) and horizontal axis (${\Omega }_p$, ${\Omega }_s$, and the transition band).
  2. Write out the mathematical definitions of passband ripple ${\alpha }_p$ (dB) and minimum stopband attenuation ${\alpha }_s$ (dB).
  3. Define the terms: Passband Edge, Stopband Edge, Passband Ripple, and Stopband Attenuation.

7.2 PYQ 2023 Question 4a [04 Marks]

Question: What is filtering? What are the applications of it?

• Answer Plan:
  1. Define filtering as a frequency-domain operation designed to select certain bands while blocking others.
  2. State at least four practical applications with keywords (Noise suppression, ECG signal cleanup, FDM Demux, demodulation, anti-aliasing).

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8. Self-Check Before Moving On

• Can you derive the impulse response of an ideal LPF from its frequency domain rect function?
• Do you know why ideal filters are mathematically non-causal?
• Can you write down the algebraic formulas linking decibel parameters (${\alpha }_p,{\alpha }_s$) with peak ripple values (${\delta }_p,{\delta }_s$)?
• Can you draw the practical low-pass tolerance curve from memory?

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Source: (k.Deergha Rao) signals and systems.pdf (Chapter 5), Azmat Sir-2309008.pdf (Lecture Slides), signal checklist.md.