Classifying Ideal Filter Magnitude Responses

Hello. In the previous lesson, you learned to locate passbands, stopbands, and cutoff frequencies on a real magnitude-response curve, using the level $3.01\text{dB}$ below the passband reference. You also saw that one cutoff separates low and high frequency regions, whereas two cutoffs bound a finite interval.

Now we make the key visual classification: given an ideal magnitude response, identify whether it is low-pass, high-pass, bandpass, or bandreject. The classification depends on one question: which frequency region is allowed to retain signal magnitude?

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Ideal responses: read the passed region first

An ideal filter has a deliberately simplified response. It passes signals perfectly in its passband and removes them completely in its stopband. If the response is normalized to unity gain, this is written as

$$|H(j\omega )|=1$$

in the passband, and

$$|H(j\omega )|=0$$

in the stopband.

The vertical edges in an ideal plot represent an instantaneous change at a cutoff frequency. Physical passive filters cannot make that instantaneous transition: their magnitude changes smoothly, and the conventional cutoff occurs at the half-power level. Nevertheless, ideal plots are valuable because they make the intended filtering action unmistakable.

The figure gives a compact visual rule:

• Pass on the left side of the graph: low-pass.
• Pass on the right side: high-pass.
• Pass in the middle: bandpass.
• Stop in the middle: bandreject.

The labels $f_L$ and $f_H$ denote lower and upper boundary frequencies. In a one-cutoff filter, the boundary is usually written $f_c$. In a two-boundary filter,

$$f_L<f_H$$

and the frequencies between them form a bounded band.

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The one-cutoff classes: low-pass and high-pass

A low-pass filter passes frequencies from DC up to its cutoff, while attenuating frequencies above that cutoff. “Low” refers to the location of the passband, not to the frequency being removed.

For an ideal low-pass filter with cutoff $f_c$,

$$|H(f)|=\{\begin{array}{ll}1,& 0\le f<f_c\\ 0,& f>f_c\end{array}$$

Its visual signature is a response that is high at the left of the graph and low at the right.

A high-pass filter does the reverse: it attenuates low frequencies and passes frequencies above its cutoff. Its ideal response is

$$|H(f)|=\{\begin{array}{ll}0,& 0\le f<f_c\\ 1,& f>f_c\end{array}$$

Its visual signature is low magnitude at the left and high magnitude at the right.

A classification check based on limiting behavior

For a one-cutoff response, inspect the two extremes rather than becoming distracted by the vertical boundary.

Response at very low frequency	Response at very high frequency	Classification
Passed	Attenuated	Low-pass
Attenuated	Passed	High-pass

For example, suppose an ideal response has unit magnitude at $100\text{Hz}$, but zero magnitude at $100\text{kHz}$. It is low-pass, regardless of the numerical value of its cutoff. Conversely, if the magnitude is zero at $100\text{Hz}$ and unity at $100\text{kHz}$, it is high-pass.

This “test the two ends” habit remains useful later when you derive RC and RL transfer functions. Before doing algebra, their low- and high-frequency limits will tell you which response type you should expect.

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The two-cutoff classes: bandpass and bandreject

When a response has two boundary frequencies, classification depends on what happens between them.

A bandpass filter passes a limited middle range while attenuating frequencies below and above it. For ideal lower and upper cutoffs $f_L$ and $f_H$,

$$|H(f)|=\{\begin{array}{ll}0,& f<f_L\\ 1,& f_L<f<f_H\\ 0,& f>f_H\end{array}$$

The graph has a central plateau or, in a realistic resonant circuit, a central peak. The middle is the passband.

A bandreject filter does the opposite. It passes frequencies below $f_L$ and above $f_H$, but attenuates the intervening range:

$$|H(f)|=\{\begin{array}{ll}1,& f<f_L\\ 0,& f_L<f<f_H\\ 1,& f>f_H\end{array}$$

Its graph has a central gap or, in a real circuit, a dip. The middle is the stopband.

You will encounter several interchangeable names:

• Bandreject
• Band-stop
• Band-elimination
• Notch, particularly when the rejected band is narrow

They describe the same broad response class: a middle band is rejected and the two outer regions are passed.

A classification check based on the middle

Use this table whenever a plot has two cutoff or boundary frequencies.

Low-frequency side	Middle interval, $f_L<f<f_H$	High-frequency side	Classification
Attenuated	Passed	Attenuated	Bandpass
Passed	Attenuated	Passed	Bandreject

The distinction is therefore not the presence of two cutoffs; both types have two. It is the state of the region between them.

Consider these verbal plots:

1. A response is near zero at low frequency, rises to unity between $2\text{kHz}$ and $5\text{kHz}$, then returns to zero. This is bandpass.

2. A response is near unity below $2\text{kHz}$, drops to zero from $2\text{kHz}$ to $5\text{kHz}$, and returns to unity above $5\text{kHz}$. This is bandreject.

The same pair of frequencies can describe either type. The response height in the central interval determines the name.

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A four-step routine for any ideal response

When you are shown an unfamiliar ideal magnitude plot, use this routine.

1. Read the horizontal axis.
   Confirm that it represents frequency, usually $f$ in hertz or $\omega$ in radians per second.

2. Locate every high-magnitude region.
   Those are the passed frequency ranges. In a normalized ideal plot, they are usually shown at $|H|=1$ or $0\text{dB}$.

3. Determine whether the response has one boundary or two.
   One boundary separates a low-frequency region from a high-frequency region. Two boundaries isolate a middle interval.

4. Name the response from the passed region.

Passed region	Filter name
Low frequencies only	Low-pass
High frequencies only	High-pass
A finite middle-frequency range only	Bandpass
Both low and high frequencies, but not the middle	Bandreject / band-stop

This procedure avoids a frequent wording error: do not classify a filter by the frequencies it suppresses unless you explicitly translate that statement into the standard name. A filter that “removes high frequencies” is low-pass, because it passes the low ones. A filter that “removes a narrow middle frequency interval” is bandreject, because that middle band is rejected.

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Ideal boundaries versus real cutoffs

It is useful to keep two representations separate.

Feature	Ideal response	Real passive response
Passband level	Exactly constant	Approximately constant or gently varying
Stopband level	Exactly zero	Finite attenuation, often increasing with frequency separation
Change at boundary	Vertical step	Smooth transition
Cutoff interpretation	Intended pass/stop boundary	Conventionally $3.01\text{dB}$ below the passband reference

Thus, an ideal low-pass plot may draw a sharp edge at $f_c$, but there is no visible $-3\text{dB}$ point on the step itself. When you later sketch first-order RC and RL responses, that sharp edge will be replaced by a rounded curve passing through the actual $-3\text{dB}$ cutoff.

The classification does not change when you move from ideal to real responses:

• A curve that is high at low frequencies and falls as frequency rises remains low-pass.
• A curve that rises toward a high-frequency plateau remains high-pass.
• A central hump remains bandpass.
• A central notch remains bandreject.

Only the sharpness and detailed shape change.

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Key takeaways

Classify an ideal filter by locating its passband:

• Low-pass: passes low frequencies; rejects high frequencies.
• High-pass: rejects low frequencies; passes high frequencies.
• Bandpass: passes a bounded middle-frequency interval; rejects frequencies on either side.
• Bandreject: rejects a bounded middle-frequency interval; passes frequencies on either side.

A response with one boundary is either low-pass or high-pass. A response with two boundaries is either bandpass or bandreject; inspect the middle interval to distinguish them. “Band-stop,” “bandreject,” and “notch” are closely related names for central-band rejection.

Next, you will draw these four ideal responses yourself, with properly labeled axes, passbands, stopbands, and cutoff frequencies.