# Why does OFDM use cyclic-prefix while QPSK uses root-raised cosine filters?

My understanding is that BPSK/QPSK/... are often implemented with root-raised cosine (RRC) filters to reduce ISI and bandwidth.

OFDM can be understood to be BPSK/QPSK/... implemented on multiple orthogonal carriers, with some additional complications like cyclic prefixes, pilot tones, and implementation using FFT/IFFT.

From what I've read, it seems OFDM doesn't usually use RRC filters.

Why is it that BPSK/QPSK/... use RCC while OFDM doesn't use RRC and instead uses a cyclic-prefix to combat ISI? In other words, why isn't OFDM implemented the same way as BPSK/QPSK/... just with multiple orthogonal carriers?

Please let me know what I'm missing! :) Thanks.

• As far as I can tell, bandwidth in OFDM is not such a big issue because all the subcarriers are modulated much more slowly. – sellibitze Nov 22 '13 at 13:02

ISI, or intersymbol interference, means different things in the context of PSK and OFDM signals. In PSK signals the symbols almost always have tails that extend, in the time-domain, into the times of other symbols. This is what they mean by "intersymbol interference".

Unfortunately they have to do this to reduce the bandwidth of the signal. They eliminate the negative effects of intersymbol interference by very carefully shaping the symbols so that when the receiver filters them through a matched filter, the other symbols have zero output at the sampling times of the other symbols.

With OFDM signals there is no overlap in the time-domain between symbols, so there is no intersymbol interference in the PSK sense of the term. What they mean by ISI in the context of ISI is when time-delayed multi-path signals cause the time-delayed symbols to overlap with the following symbol in the non-delayed signal. The cyclic prefix is added to the symbols to give the receiver enough un-corrupted data to recover the signal.

For instance, let's say that there are 64 sub-carriers and when we FFT them we get 64 samples of time-domain data. Let's also say that we think the worst multi-path we will generally see will be a time delay of 7 samples. If we make the cyclic-prefix 8 samples long then we should always have at least 65 "clean" samples, which gives us enough to recover the symbol.

• Jim, thanks for the answer (and the many others I've also read). The "ISI" term being used in different senses -- "inherent" interference vs. "external" interference makes sense. One thing I'm still missing though -- I thought with BPSK the symbols are finite length sin-waves with start phases of either 0 or pi. The Wikipedia entry link shows that the BPSK symbols don't overlap in the time-domain. Why this discrepancy? – Dan Sandberg Feb 2 '13 at 10:06
• @DanSandberg The Wikipedia entry seems to assume a rectangular pulse shaper which is also a Nyquist impulse and therefore doesn't cause any ISI. In practical systems, however, they're not used because they (1) cannot be genereated perfectly and (2) the bandwidth is theoretically infinite. Btw, OFDM inherently (because of IDFT/DFT) uses a rectangular impulse shape that accounts for the sin(x)/x shaped subcarrier spectra shown in Jim Clays figure. – Deve Feb 2 '13 at 13:50
• ...a further note: Actually both systems - single-carrier (SC) and OFDM - are using Nyquist impulses for pulse shaping to prevent ISI in the transmit signal. ISI caused by the channel is handled differently, though: OFDM uses a cyclic prefix + one-tap equalizer, whereas in (SC) systems mostly FIR or decision feedback equalizers are used to remove ISI. – Deve Feb 2 '13 at 13:54
• @DanSandberg Deve is correct, they are using rectangular pulses, probably because it makes the concepts that they are trying to explain much clearer. Almost no professional radio would do that because it greatly increases the signal bandwidth. – Jim Clay Feb 2 '13 at 17:23
• In BPSK, a rectangular pulse can't be used because it can't be generated and uses too much bandwidth. How then does OFDM use a rectangular pulse? Seems like part of the difference between OFDM and PSK is that in PSK, the assumption is that the signal is being sampled at points in the time-domain (these points must have zero-ISI) while with OFDM entire time windows are being converted into the frequency domain and represent an instance in time. Is that right? – Dan Sandberg Feb 9 '13 at 0:47

The reason cyclic prefix is used in OFDM systems is to avoid complex equalizers in receivers.

Cyclic prefix converts linear convolution of fading channel(h) & Tx data(x) into circular convolution.

Without cyclic prefix: symbol at receiver y = h*x; *- linear convolution

• It is true, but it is not what the question is asked. The answer of Jim Clay makes the point. – AlexTP Apr 5 '17 at 16:02

Cyclic prefix provides a guard period in front of each OFDM symbol to guard against multi-path delay spread introduced by the channel, while Raised Cosine Filter is used in OFDM for pulse-shaping or smoothing the discontinuities between adjacent OFDM symbols or to reduce out-of-channel emissions in the frequency domain view.

http://www.ieee802.org/3/bn/public/jan13/montreuil_01a_0113.pdf

The length of the cyclic is directly proportional to the delay spread. In LTE systems 'normal' and 'extended' cyclic prefix is used for low and high delay spread scenarios within a LTE cell.

http://www.lte-bullets.com/LTE%20in%20Bullets%20-%20Cyclic%20Prefix.pdf

BPSK, QPSK, or 16QAM uses the RRC pulse because of its excellent spectral efficiency meaning more signals (data) transmission without interference for a given channel. Regular OFDM uses a rectangular pulse instead of the RRC for minimal complexity. However, the spectral efficiency of the OFDM is reduced by sidelobes and the cyclic prefix. Therefore, the RRC pulse (filter) or similar will be used for the OFDM spectral efficiency improvement someday when the complexity is less problem. The way to use the RRC pulse in OFDM is using the offset OFDM structure that has the real and imaginary symbol alternation in the frequency and time domain. RRC OFDM does not need the cyclic prefix and the sidelobs are very low due to the excellent RRC spectrum. Therefore, RRC OFDM may be the most spectrally efficient modulation. RRC OFDM was implemented in 2001 and patented as US7065150 years later. This type of OFDM is also called as OFDM/OQAM or Staggered OFDM in various publication. FBMC (Filter Bank Multicarrier) is also this type of OFDM and it is a strong candidate for 5G instead of the regular OFDM because of this spectral efficiency reason.

The real and imaginary symbol alternation in the time and frequency domain of OFDM