Every LTE channel bandwidth uses the same 15 kHz subcarrier spacing and the same OFDM symbol duration. So only a few numbers change from one bandwidth to another: the FFT size, the sampling rate, the number of occupied subcarriers and the CP length in samples.

Followings would be a quick cheatsheet if you are DSP engineer or FPGA engineer working in LTE PHY.

I'll start with a table of all of them, and then explain each one with the arithmetic behind it.

The page covers the following topics.

• Parameter Table for each Channel Bandwidth
• Sampling Rate
• Sampling Time
• Frquency Spacing between FFT bin
• Number of Samples for each CP and OFDM Symbol
• BandWidth/Guardband Width
• Application
• Reference

Parameter Table for each Channel Bandwidth

Let's put all the numbers side by side first. A DSP or FPGA design usually needs several of them at once, for example the sampling rate to set the ADC clock and the FFT size to size the buffers.

NOTE : Refer to 3GPP 36.211-Table 6.12-1: OFDM parameters and CPRI Specification 6.4. E-UTRA sampling rates for the background information for this table.

Three rows in the table follow from the others. The occupied subcarriers are 12 x N_RB + 1, where N_RB is the number of resource blocks and the extra one is the DC subcarrier. The guard subcarriers are the FFT size minus the occupied subcarriers. The occupied channel bandwidth is the occupied subcarriers x 15 kHz, and the DL bandwidth efficiency divides it by the channel bandwidth.

N_RB comes from 36.101 v20.0.0 Table 5.6-1, which lists 6, 15, 25, 50, 75 and 100 resource blocks for the six channel bandwidths. The FFT size and the sampling rate are not specified by 3GPP. They are the usual implementation choice, and CPRI uses the same sampling rates. The 1.4 MHz channel has the lowest efficiency, 78.2%, because its 6 resource blocks use only 1.08 MHz of the 1.4 MHz channel.

Keep the time units in mind when you read the last two rows. A slot, 0.5 ms, carries 7 OFDM symbols with the normal CP, which this page calls Short CP. A subframe, 1 ms, has two slots and carries 14 symbols.

Let's get into further details for some of the important parameters.

• Only the FFT size and the sampling rate scale : the 15 kHz subcarrier spacing and the symbol timing are the same for every bandwidth.
• Occupied subcarriers = 12 x N_RB + 1 : the extra subcarrier is DC, which the eNB does not transmit in DL.
• 7 symbols per slot, 14 per subframe : with the normal CP, which the table calls Short CP.

Sampling Rate

The sampling rate is the first number an FPGA design fixes, because it sets the ADC clock and the size of every buffer after it. In LTE it is simply the subcarrier spacing times the FFT size, as the list below shows for each bandwidth.

• 20 Mhz BW Case :  Subcarrier spacing x FFT size = 15000 (15 Khz) x 2048 = 30.72 Mhz
• 15 Mhz BW Case :  Subcarrier spacing x FFT size = 15000 (15 Khz) x 1536 = 23.04 Mhz
• 10 Mhz BW Case :  Subcarrier spacing x FFT size = 15000 (15 Khz) x 1024= 15.36 Mhz
• 5 Mhz BW Case :  Subcarrier spacing x FFT size = 15000 (15 Khz) x 512= 7.68 Mhz
• 3 Mhz BW Case :  Subcarrier spacing x FFT size = 15000 (15 Khz) x 256 = 3.84 Mhz
• 1.4 Mhz BW Case :  Subcarrier spacing x FFT size = 15000 (15 Khz) x 128 = 1.92 Mhz

The FFT size is the smallest convenient size that holds all occupied subcarriers. For 20 MHz, 1201 occupied subcarriers need at least 1201 bins, and 2048 is the next power of 2. The 15 MHz case is the exception, because 1536 is not a power of 2. Some implementations run 15 MHz with a 2048-point FFT at 30.72 MHz instead, so they keep one clock for the larger bandwidths.

30.72 MHz is also 8 x 3.84 MHz, and 3.84 MHz is the UMTS chip rate. So one reference clock can serve both LTE and UMTS in a multi-mode radio. Five of the six LTE rates are 3.84 MHz multiplied or divided by a power of 2. The exception is 23.04 MHz, which is 6 x 3.84 MHz.

A receiver does not have to run at the full rate all the time. PSS, SSS and PBCH use only the central 72 subcarriers. So a UE can first decimate to 1.92 MHz, the 1.4 MHz rate, and run the cell search on 128-point FFTs before it knows the channel bandwidth.

• Sampling rate = 15 kHz x FFT size : the only difference between the bandwidths is the FFT size.
• 30.72 MHz = 8 x 3.84 MHz : the LTE rates share a reference clock with the UMTS chip rate.
• 1.92 MHz is enough for cell search : PSS, SSS and PBCH sit in the central 72 subcarriers.

Sampling Time

The sampling time is the inverse of the sampling rate. It is the smallest time step that the baseband sees, so every timing number in the PHY ends up as a count of these samples.

• 20 Mhz BW Case :  1 sec / 30.72 Mhz = 1,000,000 us/30,720,000 Hz = 0.0326 us = 32.6 ns
• 15 Mhz BW Case :  1 sec / 23.04 Mhz = 1,000,000 us/23,040,000 Hz = 0.0434 us = 43.4 ns
• 10 Mhz BW Case :  1 sec / 15.36 Mhz = 1,000,000 us/15,360,000 Hz = 0.0652 us = 65.2 ns
• 5 Mhz BW Case :  1 sec / 7.68 Mhz = 1,000,000 us/7,680,000 Hz = 0.1302 us = 130.2 ns
• 3 Mhz BW Case :  1 sec / 3.84 Mhz = 1,000,000 us/3,840,000 Hz = 0.2604 us = 260.4 ns
• 1.4 Mhz BW Case :  1 sec / 1.92 Mhz = 1,000,000 us/1,920,000 Hz = 0.5208 us = 520.8 ns

The 20 MHz value has a name in the specification. 36.211 v19.3.0 clause 4 defines the basic time unit T_s = 1/(15000 x 2048) seconds, which is about 32.55 ns. It then expresses every length in the time domain as a multiple of T_s. For example, a radio frame is T_f = 307200 T_s = 10 ms, and a slot is 15360 T_s = 0.5 ms.

For a smaller bandwidth, one sample covers several T_s. At 10 MHz one sample is 2 T_s, and at 1.4 MHz it is 16 T_s. So a timing value given in T_s is not always a whole number of samples at a low rate. The timing advance step is 16 T_s, about 0.52 microseconds (36.213 v19.4.0 clause 4.2.3), and that is exactly one sample at 1.92 MHz.

• T_s = 32.55 ns : the 20 MHz sample is the time unit of 36.211.
• Frame = 307200 T_s, slot = 15360 T_s : every duration in the specification is counted in T_s.
• Lower rates = several T_s per sample : check that a timing value is a whole number of samples at your rate.

Frquency Spacing between FFT bin

The FFT bin spacing is the sampling rate divided by the FFT size. Every bandwidth in the list below gives the same 15 kHz. This is by design: the subcarrier spacing is fixed, and only the FFT size and the sampling rate change together.

• 20 Mhz BW Case : 30.72 MHz / 2048 = 0.015 Mhz = 15 Khz = 1 RE(Resource Element) in Frequency Domain  
• 15 Mhz BW Case : 23.04 MHz / 1536 = 0.015 Mhz = 15 Khz = 1 RE(Resource Element) in Frequency Domain  
• 10 Mhz BW Case : 15.36 MHz / 1024 = 0.015 Mhz = 15 Khz = 1 RE(Resource Element) in Frequency Domain  
• 5 Mhz BW Case : 7.68 MHz / 512 = 0.015 Mhz = 15 Khz = 1 RE(Resource Element) in Frequency Domain  
• 3 Mhz BW Case : 3.84 MHz / 256 = 0.015 Mhz = 15 Khz = 1 RE(Resource Element) in Frequency Domain  
• 1.4 Mhz BW Case : 1.92 MHz / 128 = 0.015 Mhz = 15 Khz = 1 RE(Resource Element) in Frequency Domain  

One FFT bin is one subcarrier, so one bin is the width of one RE in frequency. The useful symbol length is the inverse of the spacing, 1/15 kHz = 66.7 microseconds, for every bandwidth. That is why the illustrations in the next section give 66.7 microseconds for the useful symbol, whatever the FFT size.

The bin spacing is also the ruler for frequency error. A UE whose oscillator is 1 ppm off at a 2 GHz carrier is 2 kHz off, which is about 0.13 of one bin. That is enough to leak energy into the neighbouring subcarriers, and the Physical Layer Problem Example - Frequency Error page shows what it does to a received constellation.

• Bin spacing = sampling rate / FFT size = 15 kHz : the same for every channel bandwidth.
• Useful symbol = 1/15 kHz = 66.7 microseconds : independent of the FFT size.
• Frequency error is measured against 15 kHz : 1 ppm at 2 GHz is already 0.13 of a bin.

Number of Samples for each CP and OFDM Symbol

Following illustration shows the number of samples in each CP and OFDM symbols for 20Mhz case. I will calculate the number of samples for other BW case by taking the ratio of FFT site of 20 Mhz and the FFT size of other BW.

One slot at 20 MHz. The first CP is 160 samples, the other six are 144 samples, and each useful symbol is 2048 samples.

In short, I will list up the value of A,B,C,D for each system BW.

The same slot with symbols S0 to S6. A is the first CP, B the other CPs, C the useful symbol and D one symbol including its CP.

Followings are A,B,C values for each BW

• 20 Mhz BW Case :
• A = 160
• B = 144
• C = 2048

• 15 Mhz BW Case :
• A = 160 x (FFT size for 15 Mhz/FFT size for 20 Mhz) = 160 x (1536/2048) = 120
• B = 144 x (FFT size for 15 Mhz/FFT size for 20 Mhz) = 144 x (1536/2048) = 108
• C = 2048 x (FFT size for 15 Mhz/FFT size for 20 Mhz) = 2048 x (1536/2048) = 1536

• 10 Mhz BW Case :
• A = 160 x (FFT size for 10 Mhz/FFT size for 20 Mhz) = 160 x (1024/2048) = 80
• B = 144 x (FFT size for 10 Mhz/FFT size for 20 Mhz) = 144 x (1024/2048) = 72
• C = 2048 x (FFT size for 10 Mhz/FFT size for 20 Mhz) = 2048 x (1024/2048) = 1024

• 5 Mhz BW Case :
• A = 160 x (FFT size for 5 Mhz/FFT size for 20 Mhz) = 160 x (512/2048) = 40
• B = 144 x (FFT size for 5 Mhz/FFT size for 20 Mhz) = 144 x (512/2048) = 36
• C = 2048 x (FFT size for 5 Mhz/FFT size for 20 Mhz) = 2048 x (512/2048) = 512

• 3 Mhz BW Case :
• A = 160 x (FFT size for 3 Mhz/FFT size for 20 Mhz) = 160 x (256/2048) = 20
• B = 144 x (FFT size for 3 Mhz/FFT size for 20 Mhz) = 144 x (256/2048) = 18
• C = 2048 x (FFT size for 3 Mhz/FFT size for 20 Mhz) = 2048 x (256/2048) = 256

• 1.4 Mhz BW Case :
• A = 160 x (FFT size for 1.4 Mhz/FFT size for 20 Mhz) = 160 x (128/2048) = 10
• B = 144 x (FFT size for 1.4 Mhz/FFT size for 20 Mhz) = 144 x (128/2048) = 9
• C = 2048 x (FFT size for 1.4 Mhz/FFT size for 20 Mhz) = 2048 x (128/2048) = 128

The illustration above also labels D, the samples in one OFDM symbol including the CP. D is B + C for symbols 1 to 6, and A + C for symbol 0. The table below lists D for each bandwidth and sums one slot.

Every row sums to exactly 0.5 ms at its own sampling rate. For example, 2208 + 6 x 2192 = 15360 samples, and 15360 / 30.72 MHz = 0.5 ms. The CPs of one slot take 1024 samples at 20 MHz, and 1024 cannot be split into 7 equal whole numbers. That is why the first CP is longer: 160 + 6 x 144 = 1024.

The CP length also sets the multipath tolerance. 144 samples at 30.72 MHz is 4.69 microseconds, and light covers about 1.4 km in that time. An echo delayed by less than that stays inside the CP and causes no inter-symbol interference. The extended CP is 512 samples, or 16.7 microseconds, at 20 MHz. It gives more delay tolerance, but a slot then carries only 6 symbols.

• D = B + C, or A + C for symbol 0 : 2192 and 2208 samples at 20 MHz.
• One slot = 0.5 ms at every rate : 15360 samples at 30.72 MHz, 960 at 1.92 MHz.
• CP = 4.69 microseconds : it absorbs echoes with up to about 1.4 km of extra path.

BandWidth/Guardband Width

The FFT always has more bins than the channel uses. The unused bins at both edges form the guard band, and they leave room for the transmit filter to roll off before the channel edge.

The illustration below splits the FFT bins into three parts. The occupied part is in the middle, A in Hz and B in bins, and the guard bins C and D sit on both sides of it.

Occupied bandwidth and guard bands. B + C + D is the FFT size.

• 20 Mhz BW Case :
• A = Occupied Subcarriers (including DC)  x 15 Khz = 1201 x 15 Khz = 18015 Khz = 18.015 Mhz
• B = Occupied Subcarriers (including DC) = 1201
• C + D = Guard Subcarriers = 847

• 15 Mhz BW Case :
• A = Occupied Subcarriers (including DC)  x 15 Khz = 901 x 15 Khz = 13515 Khz = 13.515 Mhz
• B = Occupied Subcarriers (including DC) = 901
• C + D = Guard Subcarriers = 635

• 10 Mhz BW Case :
• A = Occupied Subcarriers (including DC)  x 15 Khz = 601 x 15 Khz = 9015 Khz = 9.015 Mhz
• B = Occupied Subcarriers (including DC) = 601
• C + D = Guard Subcarriers = 423

• 5 Mhz BW Case :
• A = Occupied Subcarriers (including DC)  x 15 Khz = 301 x 15 Khz = 4515 Khz = 4.515 Mhz
• B = Occupied Subcarriers (including DC) = 301
• C + D = Guard Subcarriers = 211

• 3 Mhz BW Case :
• A = Occupied Subcarriers (including DC)  x 15 Khz = 181 x 15 Khz = 2715 Khz = 2.715 Mhz
• B = Occupied Subcarriers (including DC) = 181
• C + D = Guard Subcarriers = 75

• 1.4 Mhz BW Case :
• A = Occupied Subcarriers (including DC)  x 15 Khz = 73 x 15 Khz = 1095 Khz = 1.095 Mhz
• B = Occupied Subcarriers (including DC) = 73
• C + D = Guard Subcarriers = 55

The occupied subcarriers follow 12 x N_RB + 1. For 3 MHz this is 12 x 15 + 1 = 181, and for 1.4 MHz it is 12 x 6 + 1 = 73. These are the same values as in the table at the top of the page.

The guard subcarriers cannot split evenly when their count is odd. Take the 20 MHz case with the DC subcarrier at bin 0. The occupied bins run from -600 to 600, so bins -1024 to -601 give 424 guard bins on one side, and bins 601 to 1023 give 423 on the other.

The channel bandwidth is also larger than A. A 20 MHz channel has 18.015 MHz occupied, so about 1 MHz on each side is left for filtering. 36.101 v20.0.0 Figure 5.6-1 draws the same split between the channel bandwidth and the transmission bandwidth configuration. It also notes that the centre subcarrier, DC in baseband, is not transmitted in downlink.

• B = 12 x N_RB + 1 : 181 at 3 MHz and 73 at 1.4 MHz.
• C + D = FFT size - B : with an odd count, one side gets one bin more.
• Channel bandwidth is larger than A : about 1 MHz on each side at 20 MHz is left for filtering.

Application

These kind of information may sound too much detail, but they are the most basic information required for digital signal processing at the first step processing of LTE PHY signal. Examples on sharetechnote are

• LTE Physical Layer Signal : PSS
• Physical Layer Problem Example - Frequency Error
• Physical Layer Problem Example - Timing Error

Here is how a simple DL receiver uses these numbers. The receiver samples at the rate for the channel bandwidth and finds the slot boundary. It then removes A samples of CP before symbol 0 and B samples before the other symbols, and runs a C-point FFT on what is left.

The FFT output then maps to subcarriers. Bin 0 is DC. For 20 MHz, bins 1 to 600 carry the upper 600 subcarriers, and bins 1448 to 2047 carry the lower 600. The bins in between are guard, and the receiver discards them.

A mistake in one of these numbers is easy to see. A wrong CP length moves the FFT window, which looks like a timing error and rotates the phase across the subcarriers. A wrong sampling clock stretches the subcarrier spacing, which looks like a frequency error. The two PHY problem pages in the list above show both effects.

• Remove the CP, then FFT : A samples before symbol 0, B before the others, then a C-point FFT.
• Bin 0 is DC : the occupied subcarriers sit at both ends of the FFT output.
• Wrong CP or wrong clock : appears as a timing error or a frequency error.

Reference

• 3GPP TS 36.211 v19.3.0 - clause 4, Frame structure, and clause 6.12, OFDM baseband signal generation
• 3GPP TS 36.101 v20.0.0 - Table 5.6-1 and Figure 5.6-1, Transmission bandwidth configuration
• 3GPP TS 36.213 v19.4.0 - clause 4.2.3, Transmission timing adjustments