Introduction

Engineers of many backgrounds and roles are concerned with aspects around time and frequency. Often an analysis will require directly examining waveforms in one domain (e.g. time), while considering tradeoffs and impacts to related critical factors that are defined in the other domain (e.g. frequency). For instance, an electrical engineer working on a high-speed, 12 GHz bandwidth digital-to-analog converter might need to achieve a rise time of 41 picoseconds, roughly defined as the time for a signal to transition from 10% to 90% of its amplitude. This post explores how these parameters relate and considerations surrounding their measurement. But first, let's briefly review time-varying signals and their fundamental properties.

Periodic Signals

A basic sine wave is one of the simplest forms of a time-varying signal. Consider Plot 1, which shows a sine wave oscillating at a set interval. The waveform shown completes a cycle every millisecond, with this interval being defined as the period (T), measured in time. The associated fundamental frequency of this sinusoidal signal is the inverse of its period, or 1 / T. For our signal, this comes out to a frequency of 1kHz.

Fourier Analysis & Square Wave Formation

We have just seen a basic example of how frequency and time are connected, but their relationship is significantly more profound. This subject has long been explored by great minds in science and mathematics (Euler passed in 1783!) and continues to be examined with rigor through studies in complex analysis. Fourier analysis, a key mathematical tool in this field, tells us that any time varying waveform can be broken down into its basic constituents - sines and cosines. This fundamental insight, along with the inherent reciprocity of the underlying mathematics, leads us to an additional conclusion: we can not only analyze complex waveforms by breaking them down into simpler parts, but also construct new waveforms by methodically defining their constituent parts and summing them together. With this knowledge in hand, let's create a square wave via adding more sine waves to the one we have now.

The function for our basic sine wave is described by f(t) = A⋅ sin( 2 π f ⋅ t ), where A represents the amplitude and f the frequency. To simplify our discussion, we're omitting the phase term. Our next step involves adding additional sinusoids to this fundamental wave, focusing on odd harmonics while skipping the even ones. Each harmonic's amplitude is set inversely proportional to its order [e.g. the amplitude of the 3rd harmonic is 1/3, represented by 1 / 3 sin ( 2 π 3 f ⋅ t), the 5th harmonic is 1 / 5, and so on]. Participate in this experiment interactively using the + and - odd harmonic buttons between Plot 2a and Plot 2b . As you continue to add harmonics by pressing the '+ odd harmonic' button, a square wave will begin to form. However, upon a close look we will see this is not quite an ideal square wave with undershoot, overshoot, and ringing characteristics present on the waveform. This phenomenon, known as Gibbs Phenomenon, shows that it is impossible for band-limited systems to achieve an instantaneous transition from one signal level to another, or put another way, achieving an ideal square wave would require infinite bandwidth.

Bandwidth

The Fast Fourier Transform (FFT) in Plot 2b displays our signal's frequency domain spectrum, highlighting the magnitude of the signal versus frequency. With just the basic sinusoid as in our default state, there's a single line at the fundamental frequency. Notice that adding more harmonics introduces more lines at odd multiples of the fundamental frequency.

In our square wave examples, bandwidth is defined as the frequency difference between the fundamental and the highest harmonic. For example, adding two odd harmonics from the default state results in a bandwidth of 4000 Hz (5000 Hz - 1000 Hz). Thus, adding harmonics increases our bandwidth and affects the characteristics of our time domain signal as well. Let's zoom in and view more detail.

Rise Time

Now that we have familiarized ourselves with square waves within bandwidth-limited systems (i.e. all systems that can be physically realized) let's turn our attention to the rise time (RT) characteristics of these waveforms. Rise time is a crucial aspect in many electrical engineering designs. The following plot specifically focuses on the rise time segment of our signal. Rise time is typically defined as the duration required for a signal to transition from 10% to 90% of its rising edge. Note that the preshoot, overshoot, ringing, and any droop are separate attributes of the signal and the rise time measurement thresholds [10%, 90%] are assigned independently of these factors.

Bandwidth's Relationship to Rise Time

Plot 3 allows us to directly observe that BW and RT are inversely related to one another, furthering our intuition around how time and frequency are connected. The annotated result of BW * RT is a common topic of interest and some of you reading this may be saying "Hey wait a second, I've read the rule of thumb is BW * RT = 0.35, why is this waveform resulting in values greater than 0.4 at higher bandwidths?" This shows if we were solely relying on the bandwidth and this rule of thumb to calculate the rise time, or vice versa, our result would be off by a significant percentage. Work requiring precision benefits from fully understanding this concept and knowing when the rules of thumb are applicable and when sounder methods are necessary. An additional class of signals will now be explored, hinting at how the signal shape itself further impacts the relationship between bandwidth and rise time.

Waveform Impacts

Consider a Gaussian pulse train, each pulse rising and falling following a Gaussian curve with negligible overshoot. The time domain representation of this Gaussian pulse ( Plot 4a ) and its rise time measurement ( Plot 4b) demonstrate how the waveform's shape significantly influences the relationship between bandwidth and rise time.

Notice how our new Guassian signal's shape impacts the relation between bandwidth (BW) and rise time (RT). In this case we are using a spectral measurement on the Fourier transform of the time domain waveform to measure the bandwidth of the signal directly.

The BW * RT = 0.34 result indicates that for a system with a given bandwidth, the rise time associated with a Gaussian signal in this system will be faster than that associated with our square wave above. The results also indicate that a square wave with a given rise time will have greater levels of energy towards the high end of the spectrum than that of a Gaussian pulse with the same rise time. I have witnessed design changes as a result of failures in regulator emissions tests due to an electrical component acting like an antenna for a coupled switching signal, resulting in a spur on our transmitter output. This signal needed to be changed to one with less troublesome spectral content to remove the spur and resolve the issue.

Conclusion & Additional System Impacts

This topic is as relevant to the devices receiving signals as it is to the devices generating them. For example, if we need to measure a fast transient signal accurately, we will also need access to suitably high bandwidth test equipment. Otherwise, our testing will be limited to merely measuring the step response of the front end of the test equipment itself. The exact relationship between the output signal and the input signal will heavily depend on the signal under test, the equipment being used to make the measurement, and the interface between them. Achieving accurate, reliable results necessitates applying the knowledge reviewed above (and often much more) to every system component. Characteristics of both the signal and the available test equipment can heavily detriment the depicted result if misunderstood or ignored. Another common rule of thumb on this topic is that the rise time of the waveform portrayed on your oscilloscope will be the root sum of squares (RSS) of the rise time associated to the oscilloscope's step response and the rise time of the signal itself. Like the rule above relating BW to RT, this rule of thumb has limitations in applicability, and becomes a much bigger topic of concern when the test equipment's rise time is on the same order as the rise time of the signal under test. We can save that for another day though! Hopefully the post above helped build some intuition on how time, frequency, bandwidth, and rise time relate. As always, feel free to send me an email with any comments or questions.