In this post, we will discuss everything about the cyclic prefix. In order to understand cyclic prefix let’s first revise our concepts about OFDM and signal processing.

When a device transmits a signal, it may or may not reach directly to the receiver as it goes over a wire. The receiver receives multiple copies of the transmitted signal; all these copies may not be received simultaneously. In order to understand this let’s consider the scenario below:

From the above scenario, the receiver receives 3 copies of the transmitted signal. All 3 copies of the signal travel 3 different paths (paths A, B, and C). The distance each signal copy travels before they reach the receiver is different. For the above scenario let’s assume path A > path C > path B.

Assuming an OFDM pulse was transmitted wherein the first time period (call it 1st OFDM symbol S1) carries information ‘x’ and in the 2nd time period (call it 2nd OFDM symbol S2) carries information ‘y’. The following image shows the transmission of the OFDM symbols S1 & S2 along with their paths A, B, & C:

Since path A and path C are longer than path B, the received signals taking those paths will be delayed compared to the signal that traveled path B. Since path A is the longest path, it will arrive last. As mentioned in the diagram below, the difference between path A (black) and path C (red) is **Tc** seconds. Now, these copies of the signal (blue & red) carrying S1 overlaps with the signal carrying S2. This overlap is called Inter Symbol Interference (ISI).** Delayed copies of the first symbol/signal carrying S1 interfered with the symbol/signal carrying S2**. The orthogonality of the signals will be partially lost. This interference corrupts the last samples of S1 and the initial samples of S2. Following is the illustration:

So what is the solution?

**Solution 1: **Zero padding at the start of signal S2. Just after transmitting symbol S1, for the time Tc seconds, we just transmit zeros and then transmit symbol S2 as follows:

This is a good solution to avoid Inter Symbol Interference without wasting any additional signal power, but there are two main disadvantages of this:

- Zero padding introduces more noise since the residual part that is added to the beginning of the symbol at the receiver also includes independent noise terms, which degrades the signal-to-noise ratio (SNR) of the signal to be demodulated.
- Transmit waveform modifications such as time domain windowing can not be implemented with zero padding.
- We can’t use the single-tap equalizer in the frequency domain to remove the effect of the multi-path channel. One of the biggest advantages of OFDM is the single-tap equalizer which will be lost.

**Solution 2:**** **Addition of Cyclic Prefix. We add the last samples of the same symbol to its front. By adding the extra samples we lose energy. Though random samples can also be added, adding cyclic prefixes has its own advantage:

- Converting the linear convolution of signal and channel to circular convolution. As per the Fourier transform property: The linear convolution of two signals in time is the same as the multiplication of those signals in the frequency domain. Therefore at the receiver side received signal is nothing but a multiplication of the transmitted signal with the wireless channel (in frequency) with added AWGN noise. Therefore a simple single-tap equalizer can be used to remove the channel effect from the transmitted signal. If there is no circular convolution involved then there will be ICI (Inter Carrier Interference).
- Since the cyclic prefix is nothing but the samples from the same signal, it can also be used in time synchronization.

Zero padding or the addition of a cyclic prefix reduces the overall system capacity. If Tc increase compared to T then overhead increases therefore data rate decrease. For our above example if the time difference between path B and path A increases then Tc increases therefore data rate decreases. This path delay is called the delay spread of the channel. **Delay spread is the difference between the time of arrival of the earliest component of the signal (path B) and the time of arrival of the last multipath component (path A).**

Cyclic Prefix overhead for large and small cells: If the cell radius increase then the delay spread increase, the time difference between path B and path A increases, Tc increases, and the cyclic prefix increases. For small cells since the coverage area is small delay spread is also small, therefore the required cyclic prefix is also small.

4G and 5G define 2 different cyclic prefixes: normal cyclic prefix and extended cyclic prefix. The use of extended cyclic prefixes is not common it is used only for very large cells. In 5G extended cyclic prefix is defined only for 60 kHz subcarrier spacing.

**Example of the cyclic prefix from 5G:**

In 5G for 30 kHz subcarrier spacing and 100 MHz bandwidth (IFFT size 4096), cyclic prefix is defined as 320 samples for the first OFDM symbol and 288 samples for the other 13 OFDM symbols.

For this configuration, OFDM symbol time is = 1/30 kHz = 33.33 u sec. 4096 samples transmitted in 33.33 u sec.

Cyclic prefix length in time (for other symbols) = 33.33 * (288/4096) ~ 2.34 u sec.

CP overhead for the above configuration = 288/4096 = 7.03 %

Multipath supported (path difference) = C (velocity of light) * Tc (cyclic prefix time) = 3 * 10^{8} * 2.34 u ~ 703 meters. This means the distance traveled by the earliest significant component (path B) and last significant component (path A) can be 703 meters. If the distance is more than 703 meters then there will be inter-symbol interference.

**Additional notes:**

The cyclic prefix does not necessarily have to cover the entire length of the channel time dispersion. In general, there is a trade-off between the power loss due to the cyclic prefix and the signal corruption (inter-symbol and inter-subcarrier interference) due to residual time dispersion not covered by the cyclic prefix and, at a certain point, further reduction of the signal corruption due to further increase of the cyclic-prefix length will not justify the corresponding additional power loss. This also means that, although the amount of time dispersion typically increases with the cell size, beyond a certain cell size there is often no reason to increase the cyclic prefix further as the corresponding power loss due to a further increase of the cyclic prefix would have a larger negative impact, compared to the signal corruption due to the residual time dispersion not covered by the cyclic prefix.

References:

- Fundamentals of wireless communication by David Tse
- TS 38.211: Physical channels and modulation (3GPP 5G-NR reference)

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