How to understand the 5-fold rule for selecting oscilloscope bandwidth

I believe that those who often use oscilloscopes have heard of the "5x rule" of oscilloscopes. What does it mean exactly? Why is there a 5x rule instead of a 3x or 4x rule?

Today I will briefly talk to you about the origin of the 5x rule.

First, let me briefly introduce the definition of oscilloscope analog bandwidth. The entire analog channel of the oscilloscope is equivalent to a low-pass filter, and the oscilloscope bandwidth refers to the 3dB cutoff frequency of the low-pass filter. If a sine wave signal with the same frequency as the oscilloscope's calibrated bandwidth is tested, the voltage amplitude test result will drop to 0.707 of the true voltage value. If expressed in logarithmic form, the measured amplitude will be reduced by 3dB.

Figure 1. Oscilloscope bandwidth definition (BW=100MHz)

When testing the signal amplitude, if you want the test accuracy to be at least 98%, there are requirements for the analog bandwidth of the oscilloscope. Because the frequency response of the analog channel is not flat, the analog channel of the oscilloscope with BW≤1GHz ​​is usually a first-order low-pass filter frequency response with a certain roll-off characteristic, as shown in Figure 1. The higher the frequency, the greater the attenuation of the signal and the greater the amplitude test error. Therefore, high-precision amplitude testing places higher requirements on the oscilloscope bandwidth.

So how do you choose the oscilloscope bandwidth to ensure that the test accuracy is no less than 98%?

Here we only discuss the case where the oscilloscope analog channel is a first-order low-pass filter frequency response. For a first-order low-pass filter, its transfer function can be expressed as:

Where,   is the 3dB cutoff frequency of the low-pass filter.

Assuming the amplitude of the sine wave signal is 1V, when the test accuracy is not less than 98%, the following relationship needs to be satisfied:

After simplification, we can get:

The above formula shows that when testing a sine wave signal, if you want to ensure 98% amplitude test accuracy, the oscilloscope bandwidth must be no less than 4.93 times the signal frequency.

For convenience, it is generally recommended that the oscilloscope bandwidth be at least 5 times the frequency of the sine wave signal. This is the origin of the 5-fold rule.

The 5x rule has a long history. In the past, the bandwidth of oscilloscopes was not high, and oscilloscopes with bandwidths below 1 GHz were more widely used, so the 5x rule was very popular. Even now, this rule is still applicable to mid- and low-end oscilloscopes.

Nevertheless, when choosing a high-bandwidth oscilloscope, you do not need to follow the 5-fold rule. This is not because the analog channel frequency response of a high-bandwidth oscilloscope is very flat. On the contrary, it is still a low-pass filter frequency response with small low-frequency attenuation and large high-frequency attenuation. It is because high-end oscilloscopes often have DSP filter functions, as shown in the figure below, which compensates the frequency response of the analog channel in the form of a digital algorithm, thereby obtaining a flat amplitude-frequency response and a linear phase-frequency response.

DSP filter is very necessary for the acquisition of broadband signals and high-speed bit stream signals. Don’t think that this will “modify” the signal to be tested and disrupt the test. On the contrary, DSP filter will improve the signal test accuracy because it compensates for the frequency response of the oscilloscope analog channel instead of changing the signal to be tested.

Figure 2. High-end oscilloscopes support DSP filter functions to improve signal testing accuracy