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Rule of thumb: Signal bandwidth from rise time

06 Jan 2014  | Eric Bogatin

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What is the highest sine wave component in an ideal square wave? Even though they drop off rapidly and get smaller and smaller, every one is critically important to recreate the 0 psec rise time of the ideal square wave. But what if we don't need such a short rise time?

Rise time and bandwidth
To find the relationship between the rise time of a signal and its bandwidth, we are going to engineer a specific spectrum so we know exactly what the bandwidth is and measure the actual 10-90 rise time we are able to achieve for that time domain signal.

We start with the spectrum of the ideal square wave and select just the first n harmonics, with the amplitudes as they are in the ideal spectrum. The amplitudes of all the harmonics in the spectrum above n are set to 0. The highest sine wave frequency component in the spectrum is absolutely unambiguous, we engineered it to be n.

We take this frequency domain spectrum and turn it back into a time domain signal. We measure the 10-90 rise time of the resulting signal. Figure 2 shows the recreated time domain waveforms from these engineered-bandwidth spectra, for the case of including just up to the n = 1 harmonics, just up to the n = 3, just up to the n = 17 and just up to the n = 19 harmonics. For each engineered waveform, we show just the expanded view of the rising edge of the signal.

Figure 2: Recreated waveforms from engineered-bandwidths, as the highest harmonic included increases.

For each waveform, we know precisely what the bandwidth is. We engineered it to be n. We can measure the 10-90 rise time off the graph. When we compare the engineered bandwidth to the 10-90 rise time, we see a pattern, as shown in figure 3.

Figure 3: The 10-90 rise time plotted for each waveform, compared to the engineered bandwidth of the waveform.

On this plot, I drew a straight line to empirically match the bandwidth to the rise time. It is

For example, when the rise time is 1 nsec, the highest sine wave frequency component I need in the spectrum to create this 1 nsec rise time signal is 350MHz. If the rise time is 350 psec, I only need frequency components up to 1GHz to re-create the rising or falling edge of the signal.

If you are worrying about whether the 0.35 should be 0.5, don't use this simple relationship. If it is important to know whether the bandwidth of a signal is 1.3GHz or 1.5GHz, don't use the single term, bandwidth, to describe your signal. You should probably use the entire spectrum of the signal.

Now you try it:

1. What is the bandwidth of a DDR signal with a 0.3 nsec rise time?

2. What is the bandwidth of a USB signal with a 50 psec rise time?

3. What is the rise time for a signal if it has a bandwidth of 3.5GHz?

Leave your answers and your own examples in the comments section.

About the author
Eric Bogatin is Signal Integrity Evangelist at Teledyne LeCroy.
To download the PDF version of this article, click here.

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