# How to window a time domain signal for 6 symbols with cyclic prefix length of 512 samples?

Based on LTE stds. I am running a simulation for 6 symbols (each having 1200 central subcarriers) each in a 2048 FFT/IFFT bin. To these symbols are appended 512 samples as cyclic prefixes. I have to window each symbol to see the impact on OOB. How do i window them ?

• I am not sure if this is the core of your question or if there is something more specific but in case it wasn't clear, windowing is simply multiplying the time domain signal by a particular window prior to taking the FFT. Did you have a specific window in mind? It would make sense to window to measure OOB but what specifically led you to this point where you knew you had to do that (again trying to get to some more specifics on what you already know and if a particular window was specified). – Dan Boschen Dec 20 '18 at 12:06
• I have to apply Hanning, Hamming, Tukey etc windows but I am not sure what window length should I take. In this case do I specify the window length as (2048 + 512) and do a point wise mul or somethng called as overlap needs to be considered since I have six symbols Also I need to window to see how much each window lowers the OOB and the impact on BER and EVM – samz12 Dec 20 '18 at 12:19

Assuming that it would suffice to have an average result of all six symbols, I suggest that you simply concatenate all symbols with cyclic prefix in time and then window and FFT the whole dataset. This would give you the same result as an average of the result you would get for each symbol + prefix independently. Therefore the window length would be (2048+512)*6 = 15,360.

You would only consider overlap techniques when you need to continuously stream an fft process, or if you have a dataset that is not feasible to process with one fft. 15,360 is quite reasonable to do as one fft process.

Windowing involves multiplying the time domain data by the window weight sample for sample. After windowing the resulting windowed data is FFT'd. Purpose is to reduce spectral leakage from strong tones, thereby increasing the dynamic range and ability to see weaker signals together with stronger signals. This is at the expense of frequency resolution, or the ability to discern between two closely spaced tones that can be approximately the same power level (with decreased resolution, they would appear as one tone).

Note that the window will reduce the total power of the signal. For the measurement of power in bins where there are signals of similar power in adjacent bins (such as noise or waveforms that do not occupy single frequency bins), the non-coherent gain of the window would be used to compensate the measurement). For measurement of a single tone that occupies one bin (such as done with a calibration measurement), the coherent gain of the window would be used. The difference (in dB) between the coherent gain and non-coherent gain is $$10Log10(RBW)$$, where RBW is the resolution bandwidth. This makes sense as with a rectangular window (no further windowing other than selecting the data) the RBW is 1 bit, but any other window increases this bandwidth, thus if you sum square the bins to get the total power in the waveform, if you windowed other than the rectangular window you would be over-counting if there is similar power in adjacent bins since each bin includes the power in adjacent bins. The second link below goes into that further.

More on windowing:

How to apply Hamming Window?

How to calculate resolution of DFT with Hamming/Hann window?

• Thanks a ton for the explanation. I have a question though. 1. What would be the case if I were to see individual symbols in freq domain after they have been windowed individually?. – samz12 Dec 20 '18 at 12:59
• You window in the time domain, NOT the frequency domain so not sure what you are asking. My favorite window is the Kaiser window; I like having that extra "knob" called beta where I can trade-off dynamic range for resolution bandwidth directly. All the windows are easily accessible with a single command in Matlab, Octave and Python. – Dan Boschen Dec 20 '18 at 13:05
• If you mean that you have the results from someone else that windowed and then FFT'd the individual symbols, then you can get to the same result if you take the average of the (conjugate) squares for each bin, given you are interested in power spectral density (I assume). Note that you do need to pay careful attention to what the resolution bandwidth is, since you will ultimately want power units/ Hz I assume? – Dan Boschen Dec 20 '18 at 13:08
• I have an IFFTēd time domain signal which has now 6 symbols but I want to see the PSD of each symbol as a continuum now, thats what I was kinda asking? I get that I can window the entire spectrum and then observe the avg. PSD but if I wanted to see each symbol in freq domain (their PSD) after I window them in time domain, what should be the course of action? – samz12 Dec 20 '18 at 13:22
• Do you mean “FFT’d” the time domain signal? I am not sure what you mean by a continuum, but if you want to see the PSD of each then you can window each (2048+512 samples) and take the shorter duration FFT, which will give you the result for each symbol with a wider resolution bandwidth (so more total power in each bin). – Dan Boschen Dec 20 '18 at 16:53