# OFDM sensitivity to sampling time offset (with half of time interval between samples)

In simple QAM modulation symbol synchronization is crucial: the samples need to be taken (or interpolated and resampled to ) exactly corresponding to the time when the symbol was emitted.

This is why symbol syncronization methods like Muller-Muller was developed. I call this - for the sake of contrasting it with coarse OFDM symbol syncronization (achievable without interpolation and resampling) - fine-symbol-syncronization or smaller-than-time-interval-between-samples syncronizaiton.

Now, the question is, if such fine syncronization is important or not for OFDM. In OFDM there is the cyclic prefix which is much longer than the time interval between samples and there is also the FFT. It is not clear if in the case of OFDM the fine-symbol-syncronization is really neccessary. It might be that the guard interval+FFT makes it unneccessary.

One reason why fine-syncronization should not matter is that the purpose of the guard interval (cyclic prefix) is to make demodulation insensitive to channel response (multi path due to reflections).

Is it important with OFDM to carry out fine, i.e. smaller-than-time-interval-between-samples syncronization ?

EDIT: this book seems to discuss this question in Section "5.2.4 Synchronization Errors", it says that timing offset causes rotation - increasing in strength for sub-carriers further away from the center - need to read it more carefully.

## 3 Answers

Assuming frequency lock, OFDM does not need any per sample fine time synchronization or any per sample interpolation. Instead the receiver can either synchronize to the entire OFDM frame, or determine the frame time to sample time offset (which may include a smaller-than-time-interval-between-samples component) after the IFFT to post correct for any (linear, and thus very deterministic) phase rotation in each orthogonal subcarrier. With enough pilot tones or other redundancy in each OFDM frame, these corrections can even be done after initial demodulation done without any sample time or frame time synchronization (other than frequency lock and a coarse enough sync to get the correct whole frame. But this coarse sync can be off my many samples, as long as it's somewhere within the frame time including cyclic prefix minus the maximum multi-path tolerance).

But without a sufficiently accurate frequency lock of the sample rate (receiver to sender), interpolation (at much-smaller-than-time-interval-between-samples deltas) of the entire sample frame may be required before the IFFT to produce subcarrier orthogonality.

• Thanks for the answer hotpaw2 . "determine the frame time to sample time offset" - how can that be determined ? By using a known preamble ? May 19, 2016 at 8:36

Yes, fine timing synchronization is still very important for OFDM. As you know, an FFT is typically used to demodulate an OFDM signal once it's been properly synchronized and stripped of cyclic prefix. To see what would happen if you aren't finely synchronized in time, recall the time-shifting property of the DFT:

$$x[n-l] \Leftrightarrow X[k] e^{j\frac{2\pi kl}{N}}$$

That is, an arbitrary timing offset $l$ (which need not be an integer for this formula to be valid) causes a linear phase shift in each output bin that is proportional to its distance from zero frequency.

So, your subcarriers near the center of the band will not be affected that much, but as you move outward, your demodulated signals will experience an increasing amount of phase rotation. That phase rotation is related to the unknown timing error $l$, and this value will likely vary slightly from symbol period to symbol period (assuming you don't have perfect symbol rate synchronization either). This all adds up to degraded performance.

The solution? You need fine symbol synchronization in OFDM, just like with any other digital modulation format.

• would equalization allow for a time offset? If i understand it correctly, equalization should be able to correct for a phase shift. I know the OP didn't mention equalization, but I'm just wanting to make sure i'm understanding things correctly. I suppose normal QAM would benefit from equalization the same thought, wouldn't it? May 18, 2016 at 13:09
• Yes, it's possible for an equalizer to help correct for a small amount of timing error, just like it could for single-carrier modulation. However, it's best to still include proper timing recovery so the equalizer can focus on the channel effects instead. May 18, 2016 at 13:15
• Many thanks for the detailed answer Jason. How is such fine synchronization done for OFDM ? Can the Schmidl & Cox method - for example - (home.mit.bme.hu/~kollar/papers/Schmidl2.pdf) be used for fine synchronization ? May 19, 2016 at 8:04
• The Schmidl paper says "A symbol-timing error will have little effect as long as all the samples taken are within the length of the cyclically-extended OFDM symbol." on the bottom of page 2, is that incorrect ? May 19, 2016 at 8:12
• When they say "little effect" they mean the effect that I described above. It is possible to equalize it, but if the timing error is large enough, you may have to deal with additional ambiguities in your symbol decoding (if you were using QPSK and a particular subcarrier was shifted ~90 degrees due to timing error, for example). May 19, 2016 at 12:30

Although my answer is "a bit" late and the answers given earlier are not wrong, I'd like to add some details that might help the visitors.

First, the main difference between single-carrier (SC) transmission and OFDM is the sampling rate. While (without taking into account any time/frequency offsets) SC-QAM can be sampled at exactly the symbol rate ($f_\text{s}=R_\text{sym}$), the sampling rate for OFDM is $f_\text{s}=N\cdot R_\text{sym}$. This means that one OFDM symbol consists of $N$ samples plus guard interval, so sub-symbol-rate sampling is required by definition.

Let's look at the sub-tx-sample-rate sampling. Long story short: it's not needed.

As @hotpaw2 and @JasonR correctly stated, any timing offset results in a phase rotation of subcarrier symbols that is linear in frequency. A time synchronization after the sampling indeed is crucially important. Nevertheless, a correlation-based synchronization operating at normal sampling rate lets the receiver (assumed to have perfect sampling frequency synchronization) minimize the time offset to a range $$-\frac{1}{2f_\text{s}} \leq \tau_\text{off} \leq \frac{1}{2f_\text{s}}.$$

As given by @JasonR, such non-integer index offset of maximally $\pm0.5$ causes a rotation which is, for subcarriers far from the center frequency $f_\text{c}$ and close to band limit $f_\text{c}+f_\text{s}/2$, i.e. with subcarrier index around $k=\pm N/2$, $$\left|\Delta \varphi\right|_\text{max} \approx \frac{2\pi}{N}\cdot \frac{N}{2} \cdot \frac{1}{2} = \frac{\pi}{2}$$ In a well-designed OFDM system with a sufficient number of pilot subcarriers, this phase offset will be estimated by RX as part of the channel and corrected by the one-tap frequency-domain equalizer. So, having a sampling rate in the receiver higher than in the transmitter is not required.