Windowing an OFDM Signal in Time Domain

At the boundary of two OFDM symbols, pulse shaping smoothes the edges, resulting in the avoidance of spectral regrowth

Orthogonal Frequency Division Multiplexing (OFDM) has been introduced in a previous article as a technique suitable for high data-rate transmissions over a wireless channel. The two main advantages I mentioned were as follows:

  1. Simple one-tap equalization, and
  2. Ability to slice the spectrum and utilize each slice in an independent manner.

Due to these advantages, it was adopted as the preferred modulation in WiFi and 4G-LTE systems. The interesting part is that while many new waveforms were proposed to replace it in 5G NR, OFDM was still the modulation of choice for both downlink and uplink directions with some minor changes. In a later article, I will provide a comparison among major 5G waveform candidates along with their particular advantages and disadvantages.

The focus of this article is on pulse shaping in OFDM systems. This discussion heavily relies on the excellent article by Keysight Technologies in Ref. [1]. Originally, OFDM is a block processing scheme and hence there is no extra pulse shaping required after the iDFT at the Tx. Instead, a certain number of subcarriers, both on the positive and negative edges of the spectrum, are left unmodulated to cater for the decay of sinc sidelobes thus alleviating the subsequent digital and analog filtering tasks. In IEEE 802.11a standard, for example, $6$ subcarriers on the negative edge and $5$ subcarriers on the positive edge of the spectrum are assigned as null subcarriers.

Nevertheless, many systems still employ pulse shaping in OFDM systems to smooth the edges at the OFDM symbol boundaries that reduces the out of band power to meet the regulatory requirements. Recall that the pulse shape employed in a single-carrier system is a Raised Cosine in frequency domain. Therefore, from time frequency duality, we already know that the Raised Cosine pulse shape in an OFDM system must be in time domain. We have already covered the topic of pulse shaping in depth here and here and hence briefly describe this operation in the current context.

We created the Cyclic Prefix (CP) by appending the last $N_{CP}$ samples from the iFFT output to the beginning of the OFDM symbol. The total symbol duration thus becomes $N+N_{CP}$ samples. For obvious reasons, the next OFDM symbol begins with a different amplitude and phase at the boundary of the current symbol, as shown in Figure below for $N=64$ and $N_{CP}=16$. This discontinuity causes spectral regrowth that violates the spectral mask and creates interference for adjacent channels. Therefore, our task is to assemble consecutive OFDM symbols in a smooth fashion.

Discontinuity arising without pulse shaping

To introduce a smooth transition between the current and next OFDM symbol, a cyclic suffix (also known as cyclic postfix) is introduced. Just like a cyclic prefix is the last $N_{CP}$ samples in an $N$ sample symbol, a cyclic suffix is the first $N_{CS}$ samples out of $N$ that are attached at the end.

Let us create an example for $N=64$ and $N_{CP}=16$. The total number of samples in this OFDM symbol is equal to $N+N_{CP}=N_{\text{OFDM}}=80$ as in Figure above and it should remain the same after the windowing operation. Here, we assume a full length cyclic suffix of length $N_{CS} = 16$ samples to understand the concept.

When the first $16$ samples of OFDM symbol $1$ are appended in the end as a cyclic suffix, the symbol becomes $96$ samples long which violates the $N_{\text{OFDM}}=80$ sample condition of the original signal. Instead, an alternative scheme is applied as follows for which a graphical representation is illustrated in Figure below.

At the boundary of two OFDM symbols, pulse shaping smoothes the edges, resulting in the avoidance of spectral regrowth

  • Length $16$ cyclic suffix of symbol $1$ is windowed with the transition part of a Raised Cosine that rolls off from $1$ to $0$.
  • Length $16$ cyclic prefix of symbol $2$ is windowed with the transition part of the Raised Cosine that rolls on from $0$ to $1$.
  • These two segments are overlapped and summed together to generate a length $16$ portion that is attached between both symbols.

Notice that the waveform is now continuous going from one symbol to the next while the total OFDM symbol duration is still $80$.

Recall that one purpose of the cyclic prefix is to convert linear convolution with the channel into circular convolution. This circular convolution is a result of the periodicity that appears due to the cyclic prefix being the same as the last part of the symbol. However, now a full length cyclic suffix of the first symbol interferes with the cyclic prefix of the next symbol, thus leaving no room for the periodicity to appear.

The solution is to reduce the length of the window, i.e., cyclic suffix, to as less number of samples as possible. For the current example, a length $4$ cyclic suffix implies that the multipath immunity has reduced from $N_{CP}=16$ samples to $N_{CP}-N_{CS}=12$ samples. Only a maximum channel length of $N_{\text{Tap}} =12$ can now induce a single tap flat fading at each subcarrier of the Rx DFT output. This is the price we pay for the spectral smoothing of the waveform.

A very similar technique is known as Windowed-OFDM (W-OFDM) in which windowing is applied in a manner that the window overlaps with the complete OFDM symbol as well as the CP and the cyclic suffix. Moreover, the window amplitude is $-3$ dB at the start and end of the OFDM symbol. For a perfect reconstruction of the OFDM symbol, the cyclic suffix and cyclic prefix are not discarded and instead are added to the start and the end, respectively.

Please let me know in the comments section about your ideas regarding this post or any other topics for future discussions.

References


[1] OFDM Raised Cosine Windowing: http://rfmw.em.keysight.com/rfcomms/n4010a/n4010aWLAN/onlineguide/default.htm#ofdm_raised_cosine_windowing.htm

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4 Comments

  1. marek klemes

    Just window the time-domain symbols by raised-cosine windows, with the tails overlapping. You need to “leave room” for them in the time domain, including the CP already. THis is called Filtered OFDM, or FOFDM. In frequency domain, the subcarriers have RC shapes, instead of Sinc shapes, and are still orthogonal, but can have much lower sidelobes.

    Reply
    1. Qasim Chaudhari (Post author)

      Thanks for the insightful comment. Filtered-OFDM is also a term used in the literature for a CP-OFDM system with an additional lowpass filter inserted after the CP concatenation that can cause confusion.

      Reply
  2. Sumit Kumar

    What a coincidence. Since a week I was trying to understand this and just now I see your blog. I already went to the keysight website which you have referred, but cundnt understand much. But now with your explanation, it is crystal clear.
    Regards
    Sumit

    Reply
    1. Qasim Chaudhari (Post author)

      That’s interesting : )

      Reply

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There are 26 letters in English language and countless rules. The language of signal processing is simpler.

- It has only 1 letter: a sample at time 0. From there, we can build any discrete-time signal on which our 1s and 0s can be mapped.

- It has one major rule which is repeatedly employed for demapping the received signal to bits.


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