Where does bandwidth collide?

The width of a frequency band, measured in hertz, or, by extension, the rate at which a channel carries information, measured in bits per second. In Shannon's 1948 account the two are tied together: the capacity of a channel is the bandwidth times the logarithm of one plus the signal-to-noise ratio, so a wider band and a higher rate move together but are not the same quantity. Everyday usage collapses them, calling a fast connection "high-bandwidth" whether the constraint is spectral width or throughput. The referent is a physical channel and the number attached to it is an engineering measurement, not a description of anyone using the channel.[1]

Unit

Hertz, or bits per second

Recovery means

Not applicable; a property of a channel

Measured by

Spectral width of a band in hertz; channel capacity in bits per second

The limited mental resource available for attention, planning, and self-control, which Mullainathan and Shafir borrow the engineering word for in their 2013 book Scarcity. On their account a pressing lack, of money or of time, occupies the mind and leaves less capacity for everything else, so scarcity itself taxes the faculty. Mani and colleagues gave the claim a measurement in Science the same year: sugarcane farmers scored worse on fluid-intelligence and executive-control tasks before harvest, when poor, than after, when paid, and low-income shoppers primed to think about a large expense showed the same drop. Nothing here is in hertz or bits per second; the word is a metaphor, and the quantity behind it is a change in test performance.[3][2]

Unit

A person's momentary mental capacity

Recovery means

Capacity returns when the demand on it is removed

Measured by

Performance on fluid-intelligence and executive-control tasks, such as Raven's matrices and a Stroop task, before and after inducing financial concern

The range of frequencies over which a control system, a sensor, or an instrument responds usefully to its input. As Ogata sets it out, the bandwidth of a closed loop is the frequency at which the output falls to a fixed fraction of its low-frequency value, conventionally the point three decibels down, and it fixes how fast the system can track a changing command and how much high-frequency noise it lets through. Here bandwidth is again a frequency, but the referent is a system's dynamic response rather than a communication channel: a wider bandwidth means a quicker, less filtered response, with the trade-off against noise that quickness always carries.[4]

Unit

Hertz, or radians per second

Recovery means

Not applicable; a property of a system's response

Measured by

The frequency at which the closed-loop magnitude response falls to −3 dB of its low-frequency value