Ideal Magnitude-Response Diagrams for LP, HP, BP, and Bandreject Filters

Welcome back. In the previous lesson, you learned to classify an ideal response by locating its passband: low frequencies, high frequencies, a middle band, or everything except a middle band. You also distinguished an ideal vertical boundary from the smooth, $-3\text{dB}$ cutoff behavior of a physical passive filter.

This lesson turns that recognition skill into a drawing skill. By the end, you should be able to produce four clean, fully labeled ideal magnitude-response diagrams: low-pass, high-pass, bandpass, and bandreject. The aim is not artistic precision; it is to make the intended signal-selection behavior unambiguous.

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Choose the axes before drawing the response

A magnitude-response diagram plots how much of an input sinusoid appears at the output as frequency changes. The standard quantities are

$$|H(j\omega )|=\left|\frac{V_o}{V_i}\right|$$

for a linear magnitude ratio, or

$$20{\log }_{10}|H(j\omega )|$$

for magnitude in decibels.

For the ideal diagrams in this course, the most transparent choice is:

• Horizontal axis: frequency $f$, in hertz $(\text{Hz})$.
• Vertical axis: normalized magnitude $|H|$, in $\text{V/V}$ or simply “dimensionless.”
• Passband level: $|H|=1$.
• Stopband level: $|H|=0$.

A passband gain of unity means that, ideally, the output amplitude equals the input amplitude. If a problem specifies some other passband gain $A$, draw the plateau at $A$ instead of $1$.

For hand-drawn ideal response diagrams, a linear frequency axis is usually easiest because it allows you to show DC clearly at $f=0$. Later, when you sketch Bode plots for RC and RL filters, you will normally use a logarithmic frequency axis.

There are two legitimate horizontal-axis conventions:

Axis label	Meaning	Unit
$f$	ordinary frequency	$\text{Hz}$
$\omega$	angular frequency	$\text{rad/s}$

Use one, not both, on the same axis. This lesson will use $f$ and $\text{Hz}$.

The supplied reference image shows the four standard response shapes and labels important regions. Its rounded curves and marked $-3\text{dB}$ levels resemble practical filters; your ideal sketches will replace those rounded transitions with vertical steps.

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The ideal-drawing convention

An ideal filter is a visual model with an abrupt transition:

• Draw a horizontal line at $|H|=1$ wherever the signal is passed.
• Draw a horizontal line at $|H|=0$ wherever the signal is rejected.
• Join these levels with a vertical line at each cutoff frequency.
• Label the horizontal ranges “passband” and “stopband.”
• Place the cutoff label directly below its position on the frequency axis.

At a literal vertical discontinuity, the magnitude at the single exact cutoff frequency is a mathematical convention rather than a physically important value. Therefore, for ideal sketches, treat $f_c$, $f_{c1}$, and $f_{c2}$ as boundaries between regions.

This differs from a real first-order RC or RL response:

$$|H|=\frac{1}{\sqrt{2}}$$

at cutoff, which is

$$-3.01\text{dB}.$$

Do not draw a rounded $-3\text{dB}$ point on an ideal step response. You will do that when constructing real Bode-magnitude sketches in later modules.

A useful labeling rule is:

Filter class	Cutoff labels needed
Low-pass	one cutoff, $f_c$
High-pass	one cutoff, $f_c$
Bandpass	lower cutoff $f_{c1}$, upper cutoff $f_{c2}$
Bandreject	lower cutoff $f_{c1}$, upper cutoff $f_{c2}$

For the two-cutoff filters, always preserve the ordering

$$f_{c1}<f_{c2}.$$

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Draw the one-cutoff responses

Low-pass filter

Start with axes labeled $f(\text{Hz})$ and $|H|(\text{V/V})$. Mark $0$ at the left end of the frequency axis and $1$ on the vertical axis. Choose a point to the right of DC and label it $f_c$.

Draw the response at $|H|=1$ from $0$ up to $f_c$. At $f_c$, draw a vertical drop to $|H|=0$, then continue at zero for frequencies above $f_c$.

Your final labels should state:

$$\text{Passband: }0\le f<f_c$$ $$\text{Stopband: }f>f_c$$

The visual message is: the left side is passed. That is why it is low-pass.

When labeling the graph, put “passband” above or within the unity-level region and “stopband” near the zero-level region. A compact title such as Ideal low-pass magnitude response is also good practice.

High-pass filter

Use the same axes and the same unity normalization. Mark one cutoff, $f_c$, on the horizontal axis.

This time, start at $|H|=0$ near DC and continue that line until $f_c$. At $f_c$, draw a vertical rise to $|H|=1$, then continue at unity toward higher frequency.

The regions are:

$$\text{Stopband: }0\le f<f_c$$ $$\text{Passband: }f>f_c$$

The left side includes DC, so an ideal high-pass filter rejects DC. The right side is passed, which gives the filter its name.

A reliable memory cue is to read the unity-level portion of your own diagram: if the $|H|=1$ line lies left of the cutoff, it is low-pass; if it lies right, it is high-pass.

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Draw the two-cutoff responses

With two cutoff frequencies, first place $f_{c1}$ and $f_{c2}$ on the frequency axis, leaving a visible interval between them. Label that interval before drawing the response. This prevents the common error of producing a low-pass or high-pass sketch with an extra, meaningless tick mark.

Bandpass filter

For a bandpass response, the central interval is passed. Draw:

1. $|H|=0$ below $f_{c1}$.
2. A vertical rise at $f_{c1}$.
3. $|H|=1$ from $f_{c1}$ to $f_{c2}$.
4. A vertical drop at $f_{c2}$.
5. $|H|=0$ above $f_{c2}$.

Label the regions as

$$\text{Lower stopband: }f<f_{c1}$$ $$\text{Passband: }{f}_{c1}<f<f_{c2}$$ $$\text{Upper stopband: }f>f_{c2}.$$

The important visual feature is a central unity plateau. In a real series RLC bandpass circuit, that plateau becomes a rounded resonant peak, but the same two cutoff frequencies bound the passed band.

Bandreject filter

A bandreject response is the exact complement in shape: the middle interval is rejected. Draw:

1. $|H|=1$ below $f_{c1}$.
2. A vertical drop at $f_{c1}$.
3. $|H|=0$ from $f_{c1}$ to $f_{c2}$.
4. A vertical rise at $f_{c2}$.
5. $|H|=1$ above $f_{c2}$.

The labels are

$$\text{Lower passband: }f<f_{c1}$$ $$\text{Stopband: }{f}_{c1}<f<f_{c2}$$ $$\text{Upper passband: }f>f_{c2}.$$

A narrow bandreject filter is often called a notch filter. For this ideal drawing, “narrow” does not alter the labeling method; it only means that the separation between $f_{c1}$ and $f_{c2}$ is small compared with the surrounding frequency range.

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A compact drafting checklist

Use this checklist every time you are asked to “draw the ideal magnitude response.”

Item	What must appear on the diagram
Vertical axis	$
Horizontal axis	$f(\text{Hz})$, with DC or $0$ at the left
Response shape	Flat levels at $0$ and $1$, with vertical transitions
Cutoffs	$f_c$, or $f_{c1}$ and $f_{c2}$, marked below the axis
Regions	Every passband and stopband explicitly named
Filter identity	A title or clear annotation: low-pass, high-pass, bandpass, or bandreject

If the vertical axis is in decibels instead, the labeling changes as follows:

Linear-magnitude sketch	dB-magnitude sketch
Passband at $	H
Stopband at $	H
Vertical axis unit: $\text{V/V}$ or dimensionless	Vertical axis unit: $\text{dB}$

For an ideal dB sketch, it is conventional to show the stopband heading downward or to label it “ideal attenuation,” rather than attempting to place $-\infty \text{dB}$ at a finite point on the page.

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Frequent drawing errors

The diagrams are simple, but their labels carry technical meaning. Watch for these errors:

• Using “attenuation” as the vertical label while plotting increasing gain upward.
  Prefer $|H|$ or “magnitude gain (dB)” unless the sign convention for attenuation is made explicit.

• Omitting units.
  Write $f(\text{Hz})$ or $\omega (\text{rad/s})$, and write $|H|(\text{V/V})$ or magnitude $(\text{dB})$.

• Putting $-3\text{dB}$ on an ideal response.
  The $-3\text{dB}$ condition belongs to a real, smooth response. In an ideal plot, the cutoff is the vertical pass/stop boundary.

• Calling the entire span from $0$ to $f_{c2}$ the passband of a bandpass filter.
  Only the interval between the two cutoffs is passed.

• Showing only one passband on a bandreject response.
  A bandreject filter passes both the low-frequency and high-frequency outer regions.

• Treating a stopband frequency as another cutoff without a stated specification.
  In a basic ideal response, the cutoff itself separates the passband and stopband. A separate stopband-frequency label is used only when a design specification defines a required attenuation at that frequency.

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Key takeaways

A complete ideal magnitude-response sketch has three layers of information: the axes and units, the step-shaped magnitude response, and the named frequency regions and cutoff boundaries.

• Low-pass: unity at low frequency, zero above $f_c$.
• High-pass: zero at low frequency, unity above $f_c$.
• Bandpass: unity only between $f_{c1}$ and $f_{c2}$.
• Bandreject: zero only between $f_{c1}$ and $f_{c2}$.

For all four diagrams, use a horizontal frequency axis, a vertical magnitude axis, and explicit passband/stopband labels. Keep the ideal step response separate from the real $-3\text{dB}$ cutoff curves you will encounter shortly.

Next, the course begins first-order low-pass filters by deriving the series RC low-pass transfer function using impedance division.