Bandwidth & first filter interpretations

A frequency specification begins with a declared transfer quantity and a declared acceptance criterion. Typical terms are:

The threshold may be −3 dB relative to a plateau, −1 dB, a maximum ripple limit, a minimum rejection, or another application-specific value. The reference may be DC gain, midband gain, peak gain, or an explicitly specified level.

For the normalized first-order low-pass

At the characteristic frequency Ω=1,

Relative to the low-frequency plateau of unity, the amplitude-ratio level is

Therefore, if we choose the −3.0103 dB relative-amplitude criterion for this specific first-order low-pass, the bandwidth edge equals

The equality comes from the response shape plus the chosen criterion. It should not be generalized blindly to every transfer function.

An amplitude ratio of 1/√2 always corresponds to −3.0103 dB when represented as 20log₁₀ of that ratio.

A half-power statement requires more physical information. For a sinusoidal voltage across the same purely resistive impedance R,

so reducing voltage amplitude by 1/√2 reduces power by 1/2:

But a generic voltage transfer Vout/Vin compares voltages at two possibly different impedances. Calling its −3 dB point “half power” is unjustified unless the relevant power reference and impedances make that interpretation valid.

An ideal low-pass response has a low-frequency region that satisfies the pass criterion and attenuates increasingly as frequency rises. For the first-order RC low-pass, the response tends to one at DC and to zero as frequency tends to infinity.

An ideal high-pass response does the converse: it rejects DC/very low frequency and approaches its high-frequency pass level. For the ideal first-order high-pass

the −3.010 dB lower edge is Ω=1, but the ideal mathematical pass region extends to arbitrarily high frequency. There is therefore no finite upper bandwidth edge in that ideal first-order model.

A real high-pass circuit eventually acquires upper-frequency limitations from parasitics, device bandwidth, transmission-line effects, instruments or another intentional pole. Those limits belong to the fuller physical transfer function.

For a response with a finite accepted region bounded by a lower edge f_L and an upper edge f_H, once both edges are defined by the same stated criterion, the ordinary linear-frequency bandwidth is

This is common for band-pass specifications. But the edge criterion still matters: −3 dB relative to the peak, −1 dB relative to midband, or an application-specific tolerance will generally produce different f_L, f_H, and BW.

A center frequency may also be quoted, but its definition is context dependent. The arithmetic midpoint (f_L+f_H)/2 and geometric mean √(f_Lf_H) are different concepts and should not be substituted for one another without a model-based reason.

Cascade two ideal, non-loading, identical first-order low-pass sections, each with normalized response

The total transfer is

and its magnitude is

To find the overall −3.010 dB amplitude edge relative to the DC gain, impose

which gives

So the overall −3 dB bandwidth is only about 0.644 times the individual pole frequency. At Ω=1, each section is −3.010 dB and the cascade is −6.0206 dB.

The measured ratio may include much more than the nominal device:

A 50 Ω generator resistance, finite load impedance, oscilloscope/probe capacitance, cable loss, fixture resonances and finite instrument bandwidth can all shift an observed edge.

Therefore a measured bandwidth is meaningful only with the measurement plane and loading conditions recorded. De-embedding or correction is a model operation that must be stated, not silently assumed.

This directly prepares MS-02, where probe loading, generator impedance, channel bandwidth and reproducible waveform measurement become explicit measurement topics.

Suppose a measured low-pass sweep is sampled only at discrete frequencies. A reproducible extraction should state:

1. which transfer magnitude is plotted and how it is referenced;
2. the threshold, for example −3.010 dB relative to an independently estimated passband plateau;
3. which measured samples bracket the threshold crossing;
4. the interpolation method, if a crossing frequency between samples is reported;
5. frequency resolution, amplitude uncertainty and relevant calibration/loading conditions.

Reporting “cutoff = 10 kHz” from a sweep containing only 8 kHz and 12 kHz samples without stating interpolation or uncertainty overstates what was measured.

Real filters have finite transition regions and finite attenuation. “Passes low frequencies” does not mean every low-frequency component is reproduced with exactly unit magnitude and zero phase. “Rejects high frequencies” does not mean the output is mathematically zero outside one edge.

A defensible filter description therefore includes at least:

This language scales from a simple RC demonstration to RF filters and measured S-parameter specifications later in the curriculum.

1. For an ideal first-order RC low-pass, derive the −3.010 dB bandwidth in hertz.
2. Explain why |Vout/Vin|=1/√2 does not by itself prove that output power is half the input power.
3. An ideal first-order high-pass has f_L=2 kHz under a −3 dB criterion. What is its ideal upper edge?
4. A band-pass response satisfies the declared criterion from 8 MHz to 12 MHz. Compute its linear bandwidth.
5. Two identical buffered first-order low-pass stages have individual pole frequency 10 kHz. Compute the overall −3.010 dB frequency.
6. At the individual pole frequency, what is the cascade magnitude in dB?
7. Name four measurement elements that can change the observed bandwidth of a nominal device.
8. A sweep crosses a −3 dB threshold somewhere between 9 kHz and 11 kHz. What information should accompany any interpolated cutoff value?

AC-02 is now conceptually complete: single-frequency phasor solutions became H(jω), then magnitude/phase, pole-zero structure, Bode representation and defensible bandwidth/filter interpretation. The next Volume I course, MS-02, will make the measurement side explicit with oscilloscopes, probes, generators and reproducible waveform/frequency-response practice.