Filters and Frequency Response

Modified: Mar. 14, 2026

Published: Mar. 8, 2026

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Every real-world signal contains noise. The temperature reading from a thermistor has 50/60 Hz mains hum. The audio signal from a microphone has ultrasonic components that alias when sampled. The power supply rail has high-frequency switching noise. Filters let you keep the signal you want and reject everything else. This lesson teaches you to design, build, and analyze passive RC filters for the frequencies that matter in embedded systems. #AnalogElectronics #Filters #FrequencyResponse

What Is a Filter?

A filter is a circuit that selectively passes signals at certain frequencies while attenuating (reducing) signals at other frequencies. The four basic filter types are:

The RC Low-Pass Filter

The simplest and most commonly used filter in embedded electronics is the RC low-pass filter. You already encountered this in Lesson 2 when studying RC circuits.

Circuit

A resistor in series, followed by a capacitor to ground:

         [R]  10K
Vin o---/\/\/---+---o Vout
                |
              [C] 15nF
                |
               GND

  Low freq: C is open, Vout = Vin
  High freq: C is short, Vout = 0
  fc = 1/(2*pi*10K*15nF) = 1061 Hz

Cutoff Frequency

The cutoff frequency (also called the -3 dB frequency or corner frequency) is:

At the cutoff frequency, the output signal is attenuated to of the input (a 3 dB reduction in power).

Transfer Function

The ratio of output to input voltage as a function of frequency:

Roll-Off Rate

A first-order RC filter has a roll-off of -20 dB per decade (a factor of 10 in frequency). This means that for every 10x increase in frequency above , the output drops by a factor of 10. This is a gentle slope. For sharper filtering, you need higher-order filters or active filters.

Bode Plots

A Bode plot is a graph that shows a filter’s frequency response. It has two parts:

1. Magnitude plot: Output amplitude (in dB) vs frequency (log scale)
2. Phase plot: Phase shift (in degrees) vs frequency (log scale)

Reading a Bode Plot

For a first-order RC low-pass filter:

Magnitude:

• Flat at 0 dB for frequencies well below
• Drops by 3 dB at
• Falls at -20 dB/decade for frequencies above

Phase:

• Starts at 0 degrees at very low frequencies
• Reaches -45 degrees at
• Approaches -90 degrees at very high frequencies

Decibels Quick Reference

Decibels (dB) express the ratio of two power or amplitude levels on a logarithmic scale:

The RC High-Pass Filter

Swap the resistor and capacitor positions to get a high-pass filter:

         [C]  15nF
Vin o---||------+---o Vout
                |
              [R] 10K
                |
               GND

  Low freq: C is open, Vout = 0
  High freq: C is short, Vout = Vin
  fc = 1/(2*pi*10K*15nF) = 1061 Hz

Cutoff Frequency

Same formula as the low-pass:

But the behavior is opposite: frequencies below are attenuated, and frequencies above pass through.

Transfer Function

Common Uses

• AC coupling: Removing the DC component from a signal so you can amplify only the AC part. This is how oscilloscope “AC coupling” mode works.
• Removing sensor drift: A temperature sensor that drifts slowly can have its drift removed with a high-pass filter while preserving fast transient responses.
• Audio: Blocking DC offset before a speaker to prevent cone displacement.

Band-Pass Filter

A band-pass filter combines a high-pass and a low-pass filter in series. It passes frequencies between a lower cutoff () and an upper cutoff ():

V_in ── C_HP ──┬── R_HP ──── R_LP ──┬── V_out
               │                    │
              GND                  C_LP
                                    │
                                   GND

The bandwidth is the difference between the two cutoff frequencies:

The center frequency is:

Design Approach

Design the high-pass section first (to set ), then the low-pass section (to set ). The two sections should not load each other significantly. A rule of thumb: make the low-pass resistor at least 10 times larger than the high-pass resistor.

Worked Example: Audio Band-Pass

To pass audio frequencies from 100 Hz to 10 kHz:

High-pass section (): Choose :

Low-pass section (): Choose (10x the high-pass R to minimize loading):

Anti-Aliasing Filters

The Aliasing Problem

When an ADC samples a signal, it can only correctly represent frequencies below half the sampling rate (the Nyquist frequency):

If the input signal contains frequencies above , those frequencies “fold” back into the lower frequency range, creating phantom signals (aliases) that are indistinguishable from real signals.

Example: An ADC sampling at 1 kHz has . A 700 Hz noise signal in the input will appear as a 300 Hz alias in the sampled data (). No amount of digital filtering can remove this alias because it looks identical to a real 300 Hz signal.

The Solution: Anti-Aliasing Filter

Place a low-pass filter before the ADC input with a cutoff frequency at or below the Nyquist frequency. This attenuates frequencies above before they reach the ADC.

For a 1 kHz sampling rate, set to about 500 Hz or slightly below:

Choose and :

Close enough to 500 Hz. This single-pole filter provides -20 dB/decade roll-off. At the Nyquist frequency (500 Hz), the signal is attenuated by about 3 dB. At 1 kHz, it is attenuated by about 7 dB. For many applications, this is sufficient. For higher performance, use a two-stage RC filter (-40 dB/decade) or an active filter.

Second-Order Filters

A second-order filter uses two RC stages in series, providing -40 dB/decade roll-off (twice as steep as first-order):

       [R1] 3.3K    [R2] 33K
Vin o-/\/\/--+---o-/\/\/--+---o Vout
             |            |
           [C1] 100nF   [C2] 10nF
             |            |
            GND          GND

  Stage 1          Stage 2
  R2 >> R1 to reduce loading
  Roll-off: -40 dB/decade

The cutoff frequency of the cascade is not simply the individual cutoff frequency. Due to loading between stages, the effective cutoff is lower:

For accurate second-order filter design, use a Sallen-Key active filter topology (which uses an op-amp and provides precise control over the response shape). But for quick prototyping, two cascaded RC stages with the second R being 10x the first (to reduce loading) works adequately.

Butterworth vs Chebyshev

When you design active filters, you can choose the response shape:

For most embedded ADC applications, Butterworth is the standard choice because the flat passband means no signal distortion in the frequency range of interest.

Component Selection for Filters

Resistor Tolerance

For filters, use 1% metal film resistors when possible. The cutoff frequency depends directly on the resistor value, so a 5% tolerance resistor creates a 5% uncertainty in . For audio or measurement applications, this matters.

Capacitor Selection

• C0G/NP0 ceramic: Best stability, lowest distortion. Ideal for filter circuits. Available up to about 10 nF.
• X7R ceramic: Good stability, available in larger values. Acceptable for most filters.
• X5R/Y5V ceramic: Capacitance varies significantly with voltage and temperature. Avoid for filter circuits.
• Film capacitors: Excellent for audio filters. Stable and low-distortion.

Source Impedance Consideration

The filter’s input source must have a low impedance compared to the filter resistor. If the source impedance is comparable to R, it shifts the cutoff frequency. Solution: buffer the source with an op-amp voltage follower (Lesson 5) before the filter.

Practical Build: RC Low-Pass Filter

Components Needed

Circuit: Audio Low-Pass Filter

1. Choose component values For a 1 kHz cutoff: , . If you do not have 15 nF, use 10 nF () or two 10 nF in series plus a 10 nF in parallel for approximately 15 nF. Alternatively, use and for .

2. Build the filter Connect the resistor from the input to the output node. Connect the capacitor from the output node to ground.

3. Connect a signal source If you have a function generator, set it to produce a sine wave at 100 Hz, 1V amplitude. If you are using an MCU, generate a PWM signal and vary the frequency.

4. Measure the output at several frequencies Sweep the input frequency and measure the output amplitude at each point:

   • 10 Hz, 50 Hz, 100 Hz, 200 Hz, 500 Hz, 1 kHz, 2 kHz, 5 kHz, 10 kHz, 20 kHz

5. Record the results Calculate at each frequency. Below , the ratio should be close to 1. At , it should be about 0.707. Above , it drops progressively.

6. Plot the response Plot your measurements on a graph with frequency (log scale) on the horizontal axis and (or dB) on the vertical axis. You have just made your own Bode plot.

Alternative: No Function Generator

If you do not have a function generator or oscilloscope, you can still demonstrate filtering:

1. Use an MCU to generate PWM Program an Arduino or STM32 to generate a square wave at a specific frequency on a GPIO pin.

2. Build the RC filter Connect the filter between the GPIO pin and an ADC input on the same or different MCU.

3. Read the ADC at different PWM frequencies At low frequencies, the ADC will read a signal that toggles between high and low. At high frequencies (well above ), the ADC reads a nearly constant voltage (the average of the square wave, about ).

4. Listen to the difference Connect a passive buzzer after the filter. Generate a square wave at 2 kHz (a clear tone). Now put it through the 1 kHz low-pass filter. The buzzer tone becomes muffled because the harmonics that give the square wave its sharp edges are attenuated.

Measurement Table

Common Mistakes

How This Connects to Embedded Systems

Summary

Next lesson: oscillators and timing circuits. The 555 timer generates precise frequencies using the RC time constants you have been studying, and crystal oscillators provide the stable clock that drives your MCU.