The ideal digital waveform looks very simple, but real digital signals can have complex behavior in the time domain. This behavior in the time domain is reflected in the frequency domain, and one needs to define a digital signal bandwidth for use in circuit design. The bandwidth of a digital signal and the bandwidth of a circuit will determine whether a digital signal becomes distorted.
Understanding the digital signal bandwidth is also important beyond circuit design. In your PCB, dispersion in the dielectric substrate will influence how different frequency components in a signal behave as they propagate. This also causes signal distortion as a pulse is stretched and attenuated during propagation. Addressing both points requires calculating the digital signal bandwidth, which is not such a straightforward exercise. Here are some ways to understand digital signal bandwidth and how to use this in PCB design to ensure signal integrity.
To begin, the fundamental mathematical tool used for calculating the digital signal bandwidth is the Fourier transform operation. It’s important to understand that complex signal integrity simulation tools are not needed to calculate the bandwidth of a digital signal as long as you know its time-domain waveform. You can calculate the bandwidth of any signal in the time domain using a Fourier transform. Some time-domain waveforms are simple enough that you can calculate the expression for the Fourier spectrum by hand. In other cases, such as with multilevel time-domain signals with complex modulation, the Fourier transform needs to be calculated numerically.
The exact digital signal bandwidth you’ll see in the frequency domain depends on how you model your digital signals. In reality, the digitals signal bandwidth extends out to positive and negative infinity, and the bandwidth of a digital signal is not a specific number. As we’ll see, no matter how the time-domain waveform is defined, you’ll have to truncate the power spectrum of your signals at some point and take that limit as the edge of the digital signal bandwidth.
A two-level digital signal is essentially a square pulse with finite width in time. The Fourier transform of a square pulse is a complex sinc function, as shown in the graph below. Although the time-domain waveform is finite and ends at some specific time, the sinc function extends out to infinity.
A Fourier transform of a time-domain digital signal gives the digital signal bandwidth in the frequency domain. The frequency domain spectrum extends out to infinity. This graph shows the Fourier transform for a single square pulse, which is a sinc function.
The sinc function shown above is normally shown as a power spectrum or as an absolute value; the lobes in the power spectrum all lie above zero. The digital signal bandwidth in this case can be taken up to the frequency that includes ~75% of the total power in the signal. This particular frequency is called the 'knee frequency,' which will be discussed in more detail below.
For a stream of square pulses, the digital signal bandwidth is no longer a sinc function. Instead, you have a sum of sinc functions that are shifted in time and converge to a sum of sines and cosines. The standard formulation of Fourier series, which uses orthonormal sine and cosine functions, is used to represent a stream of square pulses with definite repetition frequency, edge rate, and amplitude. This gives the Fourier transform shown below. For a stream of repeating digital pulses with 50% duty cycle and infinite edge rate, the Fourier transform will be a set of specific frequencies with defined relative amplitudes.
Fourier transform of a stream of square pulses in the time domain.
Again, because the digital signal bandwidth extends out to infinity, we need to select some cutoff value and take this as the edge of the signal bandwidth. Most designers simply take the 5th harmonic, or 5x the repetition rate (i.e., the clock rate). This is fine for a signal with infinite edge rate as this harmonic will cover a decent portion of the power contained in the digital signal. However, real digital signals do not have infinite edge rate. As a result, the frequency covering up to ~75% of the power spectrum will depend on the edge rate, thus the edge rate is normally used to calculate some limiting bandwidth for a digital signal.
Because the digital signal bandwidth depends on the rise time of the signal, this motivates defining some metric for cutting off the bandwidth at some limit. A limit that is often used with two-level signals is the knee frequency, which is defined as shown below:
The knee frequency for a 2-level square wave with finite edge rate sets a definite limit on the digital signal bandwidth that is useful in many applications. The integral of the power spectrum from the repetition rate up to the knee frequency will contain ~75% of the signal power. Note that this is an arbitrary metric, but it is an easy way to set a limit on digital signal bandwidth that everyone can accept and understand.
Once you have an idea of the signal bandwidth, you can determine how the signal will respond to dispersion in the system and nonlinearities (e.g., in amplifying components). The need to design transmission line transfer functions or unique RF cavities to compensate for dispersion and losses seen by digital signals is a complex problem that requires numerical or analytical optimization methods to solve. Design tools should start including user-defined digital signal bandwidth in their simulation tools so that interconnects can be designed without field solvers.
At NWES, we’re experts at modeling the digital signal bandwidth and signal behavior in high speed PCB designs and sensitive analog designs. We know how to create a high quality, fully manufacturable PCB layout for your system. We're here to help electronics companies design modern PCBs and create cutting-edge technology. We've also partnered directly with EDA companies and advanced PCB manufacturers, and we'll make sure your next layout is fully manufacturable at scale. Contact NWES for a consultation.