High-Pass Filter on a Squarewave

Where a low-pass filter removes high-frequency harmonics and rounds the edges of a square wave, a high-pass filter does the opposite: it blocks the low-frequency content, including the DC component and the fundamental, and passes only the rapid transitions. Applied to a square wave, it produces a characteristic spike or “differentiated” shape at each edge, with the signal decaying back toward zero between transitions.

Note

What is an RC filter?

An RC filter is a passive first-order filter built from just two components: a resistor ( $R$ ) and a capacitor ( $C$ ). The capacitor’s impedance $Z_{C} = 1/ j ω C$ is frequency-dependent, it is high at low frequencies and low at high frequencies. Placing $R$ and $C$ in series and taking the output across one or the other forms a voltage divider whose ratio changes with frequency, which is the filtering effect. In the high-pass configuration the output is taken across $R$ : at low frequencies $Z_{C} ≫ R$ so most of the voltage is dropped across the capacitor and little reaches the output; at high frequencies $Z_{C} ≪ R$ so most of the voltage appears across $R$ and the signal passes through largely unattenuated.

The RC filter is a particularly good fit for this kind of illustrative example for several reasons. It is the simplest possible filter, one pole, two components, no inductors, so the mathematics is straightforward and the relationship between component values and behaviour is direct and intuitive. The single parameter $f_{c} = 1/ (2 π R C)$ fully characterises the filter, making it easy to explore different regimes simply by scaling $R$ or $C$ . It is also ubiquitous in real circuits, from decoupling networks and tone controls to anti-aliasing stages, so understanding its behaviour on a familiar waveform like a square wave has immediate practical value.

Transfer function of the RC high-pass filter

The output is taken across $Z_{R} = R$ rather than $Z_{C}$ , which is the complement of the low-pass case:

V_{o u t} (t) = V_{in} (t) (\frac{j R ω C}{j R ω C + 1})

import matplotlib.pyplot as plt
import numpy as np

def rc_high_pass_filter(f, R, C):
    omega = 2*np.pi*f
    return 1j*R*omega*C / (1.0 + 1j*R*omega*C)

Cutoff frequency

The cutoff frequency is the same expression as for the low-pass filter:

f_{c} = \frac{1}{2 π R C}

At $f_{c}$ , the filter attenuates the signal by $- 3 dB$ (output amplitude $= 1/ 2 \approx 70.7%$ of input). Below $f_{c}$ the attenuation increases at $+ 20 dB$ per decade as frequency decreases.

def rc_high_pass_cutoff(R, C):
    return 1.0/(2*np.pi*R*C)

Define the filter

R=1.0e3    # 1kOhm
C=1.0e-6   # 1µF

fc = rc_high_pass_cutoff(R, C)

print("Filter cut off is: {:7.2f} Hz".format(fc))

Filter cut off is: 159.15 Hz

Define the square wave

f = 50Hz

from scipy import signal

# Frequency of the square wave :
f = 50.0   # frequency in [Hz]
T = 1.0/f  # period of the signal in [s]

# Sample points :
samples = 2**15                     # a 2^N number of samples makes the FFT very fast
periods = 2                         # number of periods to show
delta_t = T * periods / samples     # time corresponding to one sample

t = np.linspace(0, 1/f * periods, samples, endpoint=False)

y = signal.square(2*np.pi*t*f)     # square wave, amplitude ±1 V, peak-to-peak = 2 V

plt.figure(figsize=(10,5))
plt.plot(1e3*t, y)                  # x-axis in ms
ax = plt.gca()
plt.grid(True)
plt.title("Square Wave")
plt.xlabel("time(ms)",position=(0.95,1))
plt.ylabel("signal(V)",position=(1,0.9))
plt.show()

Apply the filter to the square wave

The filter is applied in the frequency domain: the FFT of the square wave is multiplied bin-by-bin by the complex filter gain $H (f)$ , then transformed back with an IFFT. This is mathematically equivalent to convolution in the time domain but far more efficient for long signals.

from scipy.fftpack import fft, ifft, fftfreq, fftshift

# helper function :
def apply_filter_and_plot(R, C, samples, delta_t, y):
    y_out = ifft(fft(y) * rc_high_pass_filter(fftfreq(samples, delta_t), R, C))
    plt.figure(figsize=(10,5))
    plt.plot(1e3*t,np.real(y_out))
    ax = plt.gca()
    plt.grid(True)
    plt.title("Square wave with HP filter. R={:7.2f} Ohms, C={:0.2f} uF, cutoff={:0.1f} Hz".format(R, C*1e6, rc_high_pass_cutoff(R, C)))
    plt.xlabel("time(ms)",position=(0.95,1))
    plt.ylabel("signal(V)",position=(1,0.9))
    plt.show()

$f_{c} = 159 Hz$ , approximately $3 \times$ the signal frequency ( $C = 1 μ F$ ). The cutoff is above the fundamental ( $50 Hz$ ), so the fundamental and lower harmonics are significantly attenuated. The output shows clear spikes at each transition with visible exponential decay in between, the classic differentiator behaviour.

apply_filter_and_plot(R, C, samples, delta_t, y)

Square wave filtered at fc ≈ 159 Hz (C = 1µF)

$f_{c} = 1591 Hz$ , well above the signal frequency ( $C = 0.1 μ F$ ). The cutoff is now far above the fundamental and all significant harmonics, so almost all the signal content is attenuated. The output is nearly zero with only very faint spikes at the transitions.

apply_filter_and_plot(R, C/10, samples, delta_t, y)

Square wave filtered at fc ≈ 1591 Hz (C = 0.1µF)

$f_{c} = 15.9 Hz$ , below the signal frequency ( $C = 10 μ F$ ). The cutoff is now below the fundamental, so all harmonics pass with relatively little attenuation. The output closely resembles the original square wave, though the DC component is fully removed and there is a slight droop on the flat portions due to residual high-pass effect near the fundamental.

apply_filter_and_plot(R, C*10, samples, delta_t, y)

Square wave filtered at fc ≈ 15.9 Hz (C = 10µF)

High-Pass Filter on a Squarewave

Transfer function of the RC high-pass filter

Cutoff frequency

Define the filter

Define the square wave

Apply the filter to the square wave

See also: