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I have a very basic question about bandwidth of constant function $f(t) = K$.

As the Fourier transform of a constant pulse is a delta-function centered at zero frequency, the bandwidth (maximum value in the frequency domain) apparently seems to be zero as well.

My confusion basically lies in the process of generating a control signal or pulse within a laboratory environment, typically using equipment like an Arbitrary Waveform Generator (AWG). My general understanding is that the pulse should begin from $0$ and smoothly transition to some value, say $K$ in this case (if this is wrong, I am happy to be corrected). However, it appears to me that achieving the constant function $f(t) = K$ exactly in a real lab scenario would require infinite bandwidth, given that we're essentially transitioning our pulse from $0$ to $K$ instantaneously.

So, my question is about the conventional definition of bandwidth in signal processing. If bandwidth is solely related to the Fourier components of the signal pulse, does this imply that a constant pulse possesses zero bandwidth? And if so, does this characteristic make it straightforward to implement in the lab (mainly rooted on the typical notion that low-bandwidth signals are easier to implement compared to the high-bandwidth signals in the lab)?

Thank you in advance.

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3 Answers 3

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Usually, one would assume that the circuit is not in a transient in order to be able to talk about a "frequency analysis". So, assume that you have a DC source of K volts switched on since $t = -\infty$ and then...

The Fourier transform for $f(t) = K$, by definition is:

$F(f) = \int_{-\infty}^{+\infty} f(t) \exp(j2\pi f t) dt = K \int_{-\infty}^{+\infty} \exp(j2\pi f t) dt = K \delta(t)$

The equality must be understood in the distribution sense (I won't prove it, just google "why integrating complex exponential gives a delta"). So, typically one would see just a narrow line at 0 hz of $K$ volts or $\alpha \cdot K^2$ watts in a bandwith analyzer.

But the concept of "frequency analysis" could be thought in a broader sense, i. e. for non stationary signals. On most applications, one is just not interested in the transient and is enough with waiting a few seconds for the signal to estabilish. The filter design, for example, is typically divided in two complementary parts: one is the "frequency analysis" properly said for a stationary signal (how does it respond to different frequencies of periodical stationary signals), the other one is the "transient response" of the filter (how much time it takes to adapt to fast-transient changes of the signal).

If we would make a "frequency analysis" in the broader sense of the term for the transient, assuming the transient is a step, then find the Fourier transform of a pulse (or a step) and it will give you the sinc function or something like that which has an infinite bandwith, beacuse to generate an infinite slope you do need all the frequencies.

In short, an ideal constant function (or signal) has a bandwith of 0 (but has an spectral component in the origin of 0hz), and an ideal square pulse has infinite bandwith (but usually we are not interested on analyzing that beacuse we can wait enough time for the signal to vanish). Regarding an AWG, is not a problem to generate both kind of signals (the slopes of the steps are not infinite but very high). However, instead of specifing the function in the spectre which is difficult, one just have to configure the shape of the wave from some set of predefined models (square, sine, step, etc.), the amplitude and the fundamental frequency which is easier than specifing all the components of the spectre (may be impossible due to the number of components).

My guess is that you just should ignore the fact that a transient exists and think that you waited a sufficient amount of time. Hope that it was useful.

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  • $\begingroup$ Thank you for the answer. I just want to make sure: is the mathematical, precise definition of bandwidth, which is used in many important theorems in information theory (e.g. Shannon-theorem), is essentially the difference between maximum and minimum frequencies of a Fourier transform of a given pulse? $\endgroup$
    – karavan
    Commented May 6 at 5:09
  • $\begingroup$ The bandwith is the interval of frequencies when the amplitude in the spectrum (the function that results of Fourier transform) is still non zero. You could have bandwith on any signal, not necessary a pulse. However, it's easier to give sense to the concept in periodical stationary signals because that means if you have a signal receptor system you must be prepared to recieve a sine wave of the maximum frequency. In case of non periodical signals, the bandwith still exists but it's not so easy to give a physical interpretation, it's mainly for mathematical purposes. $\endgroup$
    – coal
    Commented May 7 at 14:43
  • $\begingroup$ And, while the bandwith is strictly the length of the interval of frequencies where the amplitude is non zero, since there are signals which have infinite bandwith and its real implementation is impossible, there are some other conventions to weaken the concept. One of them was given in the answer of Dan Boschen and is the Equivalent Noise Bandwith. You could also invent yourself a convenient definition, for example, the interval of frequencies where lies 90% of the power. $\endgroup$
    – coal
    Commented May 7 at 14:46
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You seem to be regarding two different types of signals as equivalent, yet they are not. Any signal can be decomposed into a sum of sinusoids, and the frequencies of those sinusoids define the bandwidth:

  1. The bandwidth of a step change signal (e.g. an instantaneous change from 0 to K) is infinite, which cannot be physically generated.
  2. The bandwidth of a constant signal (that is, one that never changes) is zero, which cannot be physically generated either as the act of generating something implies that you are changing something, although it might be approximated by outputting a constant signal and only analysing the period after the transient response has decayed to insignificant levels.

Provided a signal type is physically possible, how well it can be generated is dependent on the ability of your equipment's control system to track your setpoint and attenuate disturbances, and therefore the bandwidths of both are important. In general, control system performance increases as the frequency approaches zero.

All equipment I've encountered have allowed constant setpoints to be configured in a straightforward way, but their control performance has varied significantly.

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Yes a constant pulse indeed occupies zero bandwidth. This is consistent with the Fourier Transform of a constant in time, which is an impulse in frequency.

However that constant in time must extent as a constant over time to $+/-\infty$. This is the part that is not realizable both that it is non-causal and you don't have infinite time to complete your experiment.

For that reason we can never actually generate DC in the lab, but we can approximate it for our purposes by generating a constant over a finite time duration (rectangular pulse).

The Fourier Transform of a rectangular pulse in time is a Sinc function in frequency, and from that we get the bandwidth of your signal. The Sinc function itself extends to $+/-\infty$ in the frequency domain, but a common definition of bandwidth for that which is used is the "Equivalent Noise Bandwdith". This is the bandwidth of a brickwall filter (rectangular in the frequency domain, which is also unrealizable) that would result in the same output power if both functions were filters for white noise. These relationships for a given time duration $T$ in time are summarized below:

Rect in Time Sinc in Freq

This plot shows a causal pulse in the time domain and its resulting magnitude spectrum in the frequency domain. This is a useful graphic to commit to memory given how prolific this relationship is in Signal Processing applications. The big takeaway is a pulse that is of duration $T$ seconds in the time domain, has an equivalent noise BW of $1/T$ Hz in the frequency domain, and is a Sinc with a first null at $1/T$ away from the center. So the longer we hold our constant value in time, the closer the result gets to zero bandwidth.

This whole process of selecting a portion of the infinite duration time domain result is known as "Windowing", and in this case the simplest case of a Rectangular Window. We can use any other shape in time with the result being its Fourier Transform in frequency, resulting in other bandwidths as predicted by the Fourier Transform.

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