What is more worth? 1 MHz bandwidth at 100 MHz carrier frequency, or 10 MHz bandwidth at 1 GHz carrier? Conventional wisdom has it that higher carrier frequencies are more valuable because “there is more bandwidth there”. In this post, I will explain why that is not entirely correct.
The basic presumption of TDD/reciprocity-based Massive MIMO is that all activity, comprising the transmission of uplink pilots, uplink data and downlink data, takes place inside of a coherence interval:
At fixed mobility, in meter/second, the dimensionality of the coherence interval is proportional to the wavelength, because the Doppler spread is proportional to the carrier frequency.
In a single cell, with max-min fairness power control (for uniform quality-of-service provision), the sum-throughput of Massive MIMO can be computed analytically and is given by the following formula:
In this formula,
- = bandwidth in Hertz (split equally between uplink and downlink)
- = number of base station antennas
- = number of multiplexed terminals
- = coherence bandwidth in Hertz (independent of carrier frequency)
- = coherence time in seconds (inversely proportional to carrier frequency)
- SNR = signal-to-noise ratio (“normalized transmit power”)
- = path loss for the k:th terminal
- = constant, close to with sufficient pilot power
This formula assumes independent Rayleigh fading, but the general conclusions remain under other models.
The factor that pre-multiplies the logarithm depends on .
The pre-log factor is maximized when . The maximal value is , which is proportional to , and therefore proportional to the wavelength. Due to the multiplication $B T_c$, one can get same pre-log factor using a smaller bandwidth by instead increasing the wavelength, i.e., reducing the carrier frequency. At the same time, assuming appropriate scaling of the number of antennas, , with the number of terminals, , the quantity inside of the logarithm is a constant.
Concluding, the sum spectral efficiency (in b/s/Hz) easily can double for every doubling of the wavelength: a megahertz of bandwidth at 100 MHz carrier is ten times more worth than a megahertz of bandwidth at a 1 GHz carrier. So while there is more bandwidth available at higher carriers, the potential multiplexing gains are correspondingly smaller.
In this example,
all three setups give the same sum-throughput, however, the throughput per terminal is vastly different.