Loop Attenuation at frequency $f_1$

The gain of the low-pass filter at frequency $f$ is

$$G_a(f) = \cos(\pi fT_s).$$

After $M$ passes through the delay-line loop, a partial at frequency $f$ is subject to attenuation

$$\cos(\pi fT_s)^M.$$

Since the round-trip time in the loop is $N+1/2$ samples, the number of trips through the loop after $n$ samples ($n = tf_s$) is given by

$$M = \frac{n}{N + 1/2} = \frac{tf_s}{N + 1/2}= tf_1.$$

The attenuation factor at time $t=nT_s$ is given by

$$\alpha_f(t) \triangleq \cos(\pi fT_s)^{tf_1}.$$

That is, a partial or harmonic of frequency $f$, having an initial amplitude of $A$ at time 0, will have amplitude $A\alpha_f(t)$ at time $t$ seconds.