# What is a $BT$ (Bandwidth-Time) product with reference to modulation?

I would like to know what does Bandwidth-Time product mean. I understand that Bandwidth ($B$) = 1/Symbol Time ($T$), hence $BT = 1$.

• But how can it vary?
• What is its significance?
• e.g. When we say GFSK is GMSK when $BT = 0.5$, what does that mean exactly?

The $BT$ product is the bandwidth-symbol time product where $B$ is the $-3\textrm{ dB}$(half-power) bandwidth of the pulse/filter and $T$ is the symbol duration. For different applications you will find varying recommended values. In GSM telephony for instance, a $BT=0.3$ is recommended. In satellite communications with GMSK, for near-earth missions the CCSDS recommends a $BT = 0.25$ whilst for deep-space/interplanetary missions, the use $BT=0.5$ is recommended. You find more details in this CCSDS report on bandwidth-efficient modulations. See page 2-2 and page 2-3 for the mentioned recommended values.

What does that mean ? Let's say we have $1$ bit per symbol ($T$ then correspond to bit time). For a $BT = 1$, the pulse shaping the symbol spreads over one bit period duration. For $BT = 0.25$, the spread is over $4$ bit periods, for $BT = 0.3$ the spread is over approximately $3$ bit periods, and for $BT = 0.5$ the spread is over $2$ bit periods.

This means that a smaller $BT$ product results into higher ISI and a compact spectrum. Measures need to be taken for the introduced ISI in this case much more than in the case of a higher $BT$ product where less ISI is introduced and we have much less compact spectrum.

In GMSK, one of its properties is it maintains a constant envelope and that's because of the Gaussian pulse applied prior to modulation. The GMSK pulse can be defined as in equation $(1)$ below$^1$: $$g(t) = \frac{1}{2T}\left[Q\left(2\pi B\cdot\frac{t-\frac T2}{\sqrt{\ln(2)}}\right)-Q\left(2\pi B\cdot\frac{t+\frac T2}{\sqrt{\ln(2)}}\right)\right]\tag{1}$$

Where $Q(t)$ is the complementary cumulative distribution function defined as: $$Q(t)=\int_{t}^{\infty}\frac{1}{\sqrt{2\pi}}\exp\left(-\frac 12 x^2\right)dx\tag{2}$$ The figure below shows the pulse shapes for different values of the $BT$ product:

The spread of this pulse is inversely proportional to the $BT$ product and its peak amplitude directly proportional to the product. Again, this means that lower $BT$ results into a wider spread (over bit symbol period) and with lower peak amplitude and a high $BT$ results into a narrower spread with a higher peak amplitude. In conclusion, the pulse duration increases as the bandwidth of the pulse decreases.

Here they show a link between the $BT$ product, the filter’s $-3\textrm{ dB}$ cutoff frequency and the bit rate $R_b$ as: $$BT = \frac{f_{-3\rm dB_{\rm cutoff}}}{R_b}\tag{3}$$ You can say that the $BT$ somehow determines the degree of filtering. More on fundamentals and properties of GMSK can be found in this paper $^2$ and this paper $^3$. In both papers discussions in relation to equation $(3)$ and the variations in the eye pattern as a result of $BT$ product values are given.

You can find extra stuff here, here, and here.

$[1]$: John G. Proakis, Digital Communications, 4th Edition, McGraw-Hill, 2000.

$[2]$: A. Linz and A. Hendrickson, "Efficient implementation of an I-Q GMSK modulator," in IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 43, no. 1, pp. 14-23, Jan 1996.

$[3]$: K. Murota and K. Hirade, "GMSK Modulation for Digital Mobile Radio Telephony," in IEEE Transactions on Communications, vol. 29, no. 7, pp. 1044-1050, Jul 1981.

I have just some quick remarks regarding the questions and the answer above: First of all, I think that GFSK is GMSK when the modulation index is equal to 1/2, not the BT=0.5. There is a difference between these two parameters. The M refers to Minimum as in Minimum Shift Keying (MSK). Also, the constant envelope nature of GMSK is not due to the Gaussian pulse, it is actually thanks to the CPM (Continuous phase modulation) nature of such modulation, which has by definition a constant complexe envelope. Finally, please keep in mind that the bandwidth that defines the Gaussian pulse is different from the value of the occupied bandiwdth of the signal because GMSK is not a linear modulation (i.e. it not a M-PSK or M-PAM type signal).