# Equalizer design for spectral gain flatness

I'm trying to design an equalizer for spectral flatness over a 100MHz signal for an LTE channel emulator. A channel emulator simply multiples channel coefficients with an external signal from a signal generator in baseband and converts it to RF frequency.

The following figure shows the frequency response of the system measured with single tones(0dB power) at 100 KHz intervals over 120 MHz range with a center frequency of 3.8 GHz, measured at the RF end of the channel emulator. Our region of interest is the central 100 MHz which is shown in the following figure. In the above case, the channel is bypassed i.e. the channel coefficients simply represents unit gain and no frequency/phase translation. The unevenness in the frequency response is due to gain roll-off in the D/A converter and other RF components in the channel emulator. My questions are as follows,

1. Say I design an N tap LMS equalizer and obtain the filter coefficients. How to apply the coefficients from frequency domain equalizer to the time domain signal?

2. Does the frequency response have to be measured at a finer granularity than 100 KHz for constructing an equalizer with a smooth frequency response?

• hm, convolution property of the Fourier transform? (I must admit it's a bit unclear to me where to start explaining here; the point is that if you know LTE, then you know OFDM, and if you know OFDM, then you know the whole reason to do OFDM is to be able to point-wise equalize the channel). – Marcus Müller May 15 '18 at 7:09
• but I have a few questions for clarification: you mention multiplication with a channel matrix, but only describe a single channel. Channel matrices usually only occur in multi-antenna / multi-channel systems; and what you'd need to do is convolve in time domain (unless you're thinking all this in the frequency domain, but then we'd already be in synchronized CP-OFDM world, because you'd need to take care to not confuse linear with cyclic convolution). So, what exactly does your channel simulator do, mathematically? – Marcus Müller May 15 '18 at 7:13
• Hi @MarcusMüller Thanks for the response. I think the channel emulator is a red herring here. I've edited the question by adding that the channel is bypassed. The same applies for LTE/OFDM as well. This is simply a problem of equalizing DAC frequency response. – Naveen May 15 '18 at 15:05
• Is the question about how to translate a frequency-domain response at RF to an FIR pre-equalizer at baseband? If so, have a look at FDLS, specialized to the FIR case. If you want to do real-time updates with something like LMS, you can update the objective function from the frequency domain to get tap updates. Essentially, specify the frequency-domain integral for LS, then turn it into a matrix least squares by approximating the integral with a Riemann sum. I can be a bit more specific in an answer if this is what you're looking for. – user35336 May 17 '18 at 5:12