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I am currently reading a section in the Handbook for MRI Pulse Sequence book where it says that the time-bandwidth product for a SINC pulse is given by:

$$ T \Delta f = Z $$

where $T$ is the length of the pulse, $\Delta f$ is the bandwidth and $Z$ is the number of zero crossings of the SINC pulse. Now, my question might be very naive but it says that this is a dimensionless quantity but I do not understand why that will be:

For example, if the bandwidth is described in radians/sec, does it not have units of radians? In that case, how does it relate to the number of zero crossings, which is indeed a discrete number.

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    $\begingroup$ The trick is that $\Delta f$ is not in radians / second. It is in Hertz (cycles per second or, "per seconds"). $\endgroup$
    – Peter K. ♦
    Commented May 3, 2016 at 14:01
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    $\begingroup$ This is such an important lesson for scientists: Use $f$ for frequencies (number of events per second) and $\omega$ for angular frequencies (angle covered per second). And never ever interchange them. $\endgroup$
    – M529
    Commented May 3, 2016 at 14:11
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    $\begingroup$ @Luca The problem is a general confusion in science about the term frequency. A frequency is a measure of repeating events per second, e.g. number items produced per s by a machine, number of light pulses per s. Therefore the unit is $1/s$. Angular frequency can, of course, be calculated via $\omega = 2\pi\,f$ and is useful when some rotation is going on. However, it does not make in general sense to do this calculation, e.g. the number of items produced per s by a machine... there is no meaning behind this formula in that case. It is crucial to separate between $f$ and $\omega$. $\endgroup$
    – M529
    Commented May 3, 2016 at 15:56
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    $\begingroup$ @Luca For this reason, it is important to see that $f$ is measured in Hz, and $\omega$ in $1/s$. Hertz (Hz) should be used exclusively for frequencies, but not angular frequencies. Just because the unit is the same, it does not mean that the measure is the same. Compare this to torque (unit: Nm) and energy (unit J = Nm) - same unit, different meaning. $\endgroup$
    – M529
    Commented May 3, 2016 at 16:01
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    $\begingroup$ @Luca Certainly, neither school failed for you, nor did you fail in school. This ambiguity is an issue for many people - even scientists that use it day by day. It became some kind of lab slang to call $\omega$ frequency and use Hz as its unit. And this sticks to the people that are supposed to teach students about it... $\endgroup$
    – M529
    Commented May 3, 2016 at 17:42

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The length of the pulse, $T$, would be in units of time (for example, in seconds). The bandwidth would be in units of frequency (for example, in Hz). So

$$[T][\Delta f]=sec\frac{1}{sec}=1$$

That means that $Z$ is a dimensionless quantity. That makes sense, as $Z$ is just a natural number that describes the number of times that something happens (in this case, the number of zero crossings), and should not have units at all.

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  • $\begingroup$ Thanks for the answer and I see the point. But can I not go from Hz (cycles/sec) to radians per second by multiplying by $2 \pi$. $\endgroup$
    – Luca
    Commented May 3, 2016 at 15:00
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    $\begingroup$ The thing is that the formula you wrote is for frequency, not pulsatance (also known as "angular frequency"). They are two different magnitudes, although they are very related (remember that $\omega = 2\pi f$). Nevertheless, it is not a matter of units. I mean, it is not that $f$ can be expressed in Hz or rad/sec. The latter would be non-sense (it would be like expressing a mass of a body in meters! - saving the distances). So, the matter is it that they are two different physical magnitudes and have to be treated like that. $\endgroup$
    – Tendero
    Commented May 3, 2016 at 15:07

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