Solving for corresponding time constant

The time constant $\tau$ is the time to decay by $1/e$.

To solve for $\tau_f$, the time constant for frequency $f$,

$$\alpha_f(t) = e^{-t/\tau_f}$$

$$\ln \alpha_f(t) = -\frac{t}{\tau_f}$$

$$\tau_f = -\frac{t}{\ln \alpha_f(t)}$$

$$= -\frac{t}{tf_1 \ln\left(\cos(\pi f T_s)\right)}$$

$$= -\frac{1}{f_1\ln(\cos(\pi fT_s))}$$

$$= -\frac{\left(N+1/2\right)T_s}{\ln(\cos(\pi fT_s))} .$$