# Importance Sampling in Unreal Engine Formula Derivation (Irradiance term)

I am currently trying to understand IBL in Unreal Engine, and there are so much things that I don't understand about the formula. Unreal approximate the specular term of shading equation by doing split sum approximation.

This is how unreal pre-filter the environment map for specular-irradiance term.

float3 PrefilterEnvMap(float Roughness, float3 R )
{
float3 N = R;
float3 V = R;
float3 PrefilteredColor = 0;
const uint NumSamples = 1024;
for(uint i = 0; i < NumSamples; i++ )
{
float2 Xi = Hammersley( i, NumSamples );
float3 H = ImportanceSampleGGX( Xi, Roughness, N );
float3 L = 2 * dot  ( V, H ) * H - V;
float NoL =saturate( dot  ( N, L ) );
if ( NoL > 0 )
{
PrefilteredColor += EnvMap.SampleLevel( EnvMapSampler, L, 0 ).rgb * NoL;
TotalWeight += NoL;
}
}
return PrefilteredColor / TotalWeight;
}


My question is why does the total weight is the sum of saturate(dot(N,L)). My understanding on importance sampling is that we should divide our sample with the pdf we use to sample. In this case, from what I read at here the pdf should be

$$p_i(wm, wo) = \frac{D(wm)(wm \cdot wg)}{4|wo \cdot wm|}$$

My reference on pbr in unreal engine is here and here.