Phase shift of shifted impulse response
A time delay T is represented in frequency domain as a multiplicator. That multiplicator is exp(-2*Pi * f * T *j). You have 4 unit interval delay, so your formula is right. But its interpretation falters.

The complex exponential multiplicator is only phase shift, no affect to amplitude. The phase lag (=negative phase shift) caused by that delay is not constant, but increases linearly as the freguency increases. At frequency 1/T the phase lag is full 2*Pi. You see that easily: A sine voltage with period=T is delayed exactly one period. It means phase lag = 2*Pi (or phase shift -2*Pi). Higher frequencies lag more and lower frequencies less.

Actually your expression 4W is ok for the phase lag, if W=2* Pi *f. Frequency f must be given in proportion to the sampling frequency. In the beginning you used frequency indexes k= -10....10, so to be notationally consistent, we must write in this formula f=k/10.