# Spectral efficiency

Spectral efficiency, spectrum efficiency or bandwidth efficiency refers to the information rate that can be transmitted over a given bandwidth in a specific communication system. It is a measure of how efficiently a limited frequency spectrum is utilized by the physical layer protocol, and sometimes by the media access control (the channel access protocol).

The link spectral efficiency of a digital communication system is measured in bit/s/Hz, or, less frequently but unambiguously, in (bit/s)/Hz. It is the net bitrate (useful information rate excluding error-correcting codes) or maximum throughput divided by the bandwidth in hertz of a communication channel or a data link. Alternatively, the spectral efficiency may be measured in bit/symbol, which is equivalent to bits per channel use (bpcu), implying that the net bit rate is divided by the symbol rate (modulation rate) or line code pulse rate.

Link spectral efficiency is typically used to analyse the efficiency of a digital modulation method or line code, sometimes in combination with a forward error correction (FEC) code and other physical layer overhead. In the latter case, a "bit" refers to a user data bit; FEC overhead is always excluded.

The modulation efficiency in bit/s is the gross bitrate (including any error-correcting code) divided by the bandwidth.

Example 1: A transmission technique using one kilohertz of bandwidth to transmit 1,000 bits per second has a modulation efficiency of 1 (bit/s)/Hz.
Example 2: A V.92 modem for the telephone network can transfer 56,000 bit/s downstream and 48,000 bit/s upstream over an analog telephone network. Due to filtering in the telephone exchange, the frequency range is limited to between 300 hertz and 3,400 hertz, corresponding to a bandwidth of 3,400 − 300 = 3,100 hertz. The spectral efficiency or modulation efficiency is 56,000/3,100 = 18.1 (bit/s)/Hz downstream, and 48,000/3,100 = 15.5 (bit/s)/Hz upstream.

An upper bound for the attainable modulation efficiency is given by the Nyquist rate or Hartley's law as follows: For a signaling alphabet with M alternative symbols, each symbol represents N = log2 M bits. N is the modulation efficiency measured in bit/symbol or bpcu. In the case of baseband transmission (line coding or pulse-amplitude modulation) with a baseband bandwidth (or upper cut-off frequency) B, the symbol rate can not exceed 2B symbols/s in view to avoid intersymbol interference. Thus, the spectral efficiency can not exceed 2N (bit/s)/Hz in the baseband transmission case. In the passband transmission case, a signal with passband bandwidth W can be converted to an equivalent baseband signal (using undersampling or a superheterodyne receiver), with upper cut-off frequency W/2. If double-sideband modulation schemes such as QAM, ASK, PSK or OFDM are used, this results in a maximum symbol rate of W symbols/s, and in that the modulation efficiency can not exceed N (bit/s)/Hz. If digital single-sideband modulation is used, the passband signal with bandwidth W corresponds to a baseband message signal with baseband bandwidth W, resulting in a maximum symbol rate of 2W and an attainable modulation efficiency of 2N (bit/s)/Hz.

Example 3: A 16QAM modem has an alphabet size of M = 16 alternative symbols, with N = 4 bit/symbol or bpcu. Since QAM is a form of double sideband passband transmission, the spectral efficiency cannot exceed N = 4 (bit/s)/Hz.
Example 4: The 8VSB (8-level vestigial sideband) modulation scheme used in the ATSC digital television standard gives N=3 bit/symbol or bpcu. Since it can be described as nearly single-side band, the modulation efficiency is close to 2N = 6 (bit/s)/Hz. In practice, ATSC transfers a gross bit rate of 32 Mbit/s over a 6 MHz wide channel, resulting in a modulation efficiency of 32/6 = 5.3 (bit/s)/Hz.
Example 5: The downlink of a V.92 modem uses a pulse-amplitude modulation with 128 signal levels, resulting in N = 7 bit/symbol. Since the transmitted signal before passband filtering can be considered as baseband transmission, the spectral efficiency cannot exceed 2N = 14 (bit/s)/Hz over the full baseband channel (0 to 4 kHz). As seen above, a higher spectral efficiency is achieved if we consider the smaller passband bandwidth.

If a forward error correction code is used, the spectral efficiency is reduced from the uncoded modulation efficiency figure.

Example 6: If a forward error correction (FEC) code with code rate 1/2 is added, meaning that the encoder input bit rate is one half the encoder output rate, the spectral efficiency is 50% of the modulation efficiency. In exchange for this reduction in spectral efficiency, FEC usually reduces the bit-error rate, and typically enables operation at a lower signal to noise ratio (SNR).

An upper bound for the spectral efficiency possible without bit errors in a channel with a certain SNR, if ideal error coding and modulation is assumed, is given by the Shannon-Hartley theorem.

Example 7: If the SNR is 1, corresponding to 0 decibel, the link spectral efficiency can not exceed 1 (bit/s)/Hz for error-free detection (assuming an ideal error-correcting code) according to Shannon-Hartley regardless of the modulation and coding.

Note that the goodput (the amount of application layer useful information) is normally lower than the maximum throughput used in the above calculations, because of packet retransmissions, higher protocol layer overhead, flow control, congestion avoidance, etc. On the other hand, a data compression scheme, such as the V.44 or V.42bis compression used in telephone modems, may however give higher goodput if the transferred data is not already efficiently compressed.

The link spectral efficiency of a wireless telephony link may also be expressed as the maximum number of simultaneous calls over 1 MHz frequency spectrum in erlangs per megahertz, or E/MHz. This measure is also affected by the source coding (data compression) scheme. It may be applied to analog as well as digital transmission.

In wireless networks, the link spectral efficiency can be somewhat misleading, as larger values are not necessarily more efficient in their overall use of radio spectrum. In a wireless network, high link spectral efficiency may result in high sensitivity to co-channel interference (crosstalk), which affects the capacity. For example, in a cellular telephone network with frequency reuse, spectrum spreading and forward error correction reduce the spectral efficiency in (bit/s)/Hz but substantially lower the required signal-to-noise ratio in comparison to non-spread spectrum techniques. This can allow for much denser geographical frequency reuse that compensates for the lower link spectral efficiency, resulting in approximately the same capacity (the same number of simultaneous phone calls) over the same bandwidth, using the same number of base station transmitters. As discussed below, a more relevant measure for wireless networks would be system spectral efficiency in bit/s/Hz per unit area. However, in closed communication links such as telephone lines and cable TV networks, and in noise-limited wireless communication system where co-channel interference is not a factor, the largest link spectral efficiency that can be supported by the available SNR is generally used.

## System spectral efficiency or area spectral efficiency

In digital wireless networks, the system spectral efficiency or area spectral efficiency is typically measured in (bit/s)/Hz per unit area, in (bit/s)/Hz per cell, or in (bit/s)/Hz per site. It is a measure of the quantity of users or services that can be simultaneously supported by a limited radio frequency bandwidth in a defined geographic area. It may for example be defined as the maximum aggregated throughput or goodput, i.e. summed over all users in the system, divided by the channel bandwidth and by the covered area or number of base station sites. This measure is affected not only by the single user transmission technique, but also by multiple access schemes and radio resource management techniques utilized. It can be substantially improved by dynamic radio resource management. If it is defined as a measure of the maximum goodput, retransmissions due to co-channel interference and collisions are excluded. Higher-layer protocol overhead (above the media access control sublayer) is normally neglected.

Example 8: In a cellular system based on frequency-division multiple access (FDMA) with a fixed channel allocation (FCA) cellplan using a frequency reuse factor of 1/4, each base station has access to 1/4 of the total available frequency spectrum. Thus, the maximum possible system spectral efficiency in (bit/s)/Hz per site is 1/4 of the link spectral efficiency. Each base station may be divided into 3 cells by means of 3 sector antennas, also known as a 4/12 reuse pattern. Then each cell has access to 1/12 of the available spectrum, and the system spectral efficiency in (bit/s)/Hz per cell or (bit/s)/Hz per sector is 1/12 of the link spectral efficiency.

The system spectral efficiency of a cellular network may also be expressed as the maximum number of simultaneous phone calls per area unit over 1 MHz frequency spectrum in E/MHz per cell, E/MHz per sector, E/MHz per site, or (E/MHz)/m2. This measure is also affected by the source coding (data compression) scheme. It may be used in analog cellular networks as well.

Low link spectral efficiency in (bit/s)/Hz does not necessarily mean that an encoding scheme is inefficient from a system spectral efficiency point of view. As an example, consider Code Division Multiplexed Access (CDMA) spread spectrum, which is not a particularly spectral efficient encoding scheme when considering a single channel or single user. However, the fact that one can "layer" multiple channels on the same frequency band means that the system spectrum utilization for a multi-channel CDMA system can be very good.

Example 9: In the W-CDMA 3G cellular system, every phone call is compressed to a maximum of 8,500 bit/s (the useful bitrate), and spread out over a 5 MHz wide frequency channel. This corresponds to a link throughput of only 8,500/5,000,000 = 0.0017 (bit/s)/Hz. Let us assume that 100 simultaneous (non-silent) calls are possible in the same cell. Spread spectrum makes it possible to have as low a frequency reuse factor as 1, if each base station is divided into 3 cells by means of 3 directional sector antennas. This corresponds to a system spectrum efficiency of over 1 × 100 × 0.0017 = 0.17 (bit/s)/Hz per site, and 0.17/3 = 0.06 (bit/s)/Hz per cell or sector.

The spectral efficiency can be improved by radio resource management techniques such as efficient fixed or dynamic channel allocation, power control, link adaptation and diversity schemes.

A combined fairness measure and system spectral efficiency measure is the fairly shared spectral efficiency.

## Comparison table

Examples of predicted numerical spectral efficiency values of some common communication systems can be found in the table below. These results will not be achieved in all systems. Those further from the transmitter will not get this performance.

Spectral efficiency of common communication systems
Service Standard Launched,
year
Max. net bitrate
per carrier and
spatial stream,
R (Mbit/s)
Bandwidth
per carrier,
B (MHz)
R/B (bit/s⋅Hz)
Typical reuse factor, 1/K System spectral efficiency,
R/BK (bit/s⋅Hz per site)
SISO MIMO
1G cellular NMT 450 modem 1981 0.0012 0.025 0.45 N/A 17 0.064
1G cellular AMPS modem 1983 0.0003 0.030 0.001 N/A 17 0.0015
2G cellular GSM 1991 0.013 × 8 timeslots = 0.104 0.2 0.52 N/A 19 (​13 in 1999) 0.17 (in 1999)
2G cellular D-AMPS 1991 0.013 × 3 timeslots = 0.039 0.030 1.3 N/A 19 (​13 in 1999) 0.45 (in 1999)
2.75G cellular CDMA2000 1× voice 2000 0.0096 per phone call × 22 calls 1.2288 0.0078 per call N/A 1 0.172 (fully loaded)
2.75G cellular GSM + EDGE 2003 0.384 (typ. 0.20) 0.2 1.92 (typ. 1.00) N/A 13 0.33
2.75G cellular IS-136HS + EDGE 0.384 (typ. 0.27) 0.200 1.92 (typ. 1.35) N/A 13 0.45
3G cellular WCDMA FDD 2001 0.384 5 0.077 N/A 1 0.51
3G cellular CDMA2000 1× PD 2002 0.153 1.2288 0.125 N/A 1 0.1720 (fully loaded)
3G cellular CDMA2000 1×EV-DO Rev.A 2002 3.072 1.2288 2.5 N/A 1 1.3
Fixed WiMAX IEEE 802.16d 2004 96 20 4.8 14 1.2
3.5G cellular HSDPA 2007 21.1 5 4.22 1 4.22
4G MBWA iBurst HC-SDMA 2005 3.9 0.625 7.3  1 7.3
4G cellular LTE 2009 81.6 20 4.08 16.32 (4×4)  1 (13 at the perimeters) 16.32
4G cellular LTE-Advanced 2013 75 20 3.75 30.00 (8×8)  1 (13 at the perimeters) 30
Wi-Fi IEEE 802.11a/g 2003 54 20 2.7 N/A 13[citation needed] 0.900
Wi-Fi IEEE 802.11n 2007 72.2 (up to 150) 20 (up to 40) 3.61 (up to 3.75) Up to 15.0 (4×4, 40 MHz) 13[citation needed] 5.0 (4×4, 40 MHz)
Wi-Fi IEEE 802.11ac 2012 433.3 (up to 866.7) 80 (up to 160) 5.42 Up to 43.3 (8×8, 160 MHz) 13[citation needed] 14.4 (8×8, 160 MHz)
Wi-Fi IEEE 802.11ax 2019 600.5 (up to 1201) 80 (up to 160) 7.5 Up to 60 (8×8, 160 MHz) 13[citation needed] 20 (8×8, 160 MHz)
WiGig IEEE 802.11ad 2013 6756 2160 3 N/A 1[citation needed] 3
Trunked radio system TETRA, low FEC 1998 4 timeslots = 0.019 (0.029 without FEC) 0.025 0.8 N/A 17 0.1
Trunked radio system TETRA II with TEDS, 64-QAM, 150 kHz, low FEC 2011 4 timeslots = 0.538 0.150 (scalable to 0.025) 3.6 N/A
Digital radio DAB 1995 0.576 to 1.152 1.712 0.34 to 0.67 N/A 15 0.07 to 0.13
Digital radio DAB with SFN 1995 0.576 to 1.152 1.712 0.34 to 0.67 N/A 1 0.34 to 0.67
Digital TV DVB-T 1997 31.67 (typ. 24) 8 4.0 (typ. 3.0) N/A 17 0.57
Digital TV DVB-T with SFN 1996 31.67 (typ. 24) 8 4.0 (typ. 3.0) N/A 1 4.0 (typ. 3.0)
Digital TV DVB-T2 2009 45.5 (typ. 40) 8 5.7 (typ. 5.0) N/A 17 0.81
Digital TV DVB-T2 with SFN 2009 45.5 (typ. 40) 8 5.7 (typ. 5.0) N/A 1 5.7 (typ. 5.0)
Digital TV DVB-S 1995 33.8 for 5.1 C/N (44.4 for 7.8 C/N) 27.5 1.2 (1.6) N/A 14 0.3 (0.4)
Digital TV DVB-S2 2005 46 for 5.1 C/N (58.8 for 7.8 C/N) 30 (typ.) 1.5 (2.0) N/A 14 0.4 (0.5)
Digital TV ATSC with DTx 1996 32 19.39 1.6 N/A 1 3.23
Digital TV DVB-H 2007 5.5 to 11 8 0.68 to 1.4 N/A 15 0.14 to 0.28
Digital TV DVB-H with SFN 2007 5.5 to 11 8 0.68 to 1.4 N/A 1 0.68 to 1.4
Digital cable TV DVB-C 256-QAM mode 1994 38 6 6.33 N/A N/A N/A
Broadband CATV modem DOCSIS 3.1 QAM-4096, 25 kHz OFDM spacing, LDPC 2016 1890 192 9.84 N/A N/A N/A