The infinite impulse response is a property of a filter of digital signal processing. The filter response of the impulse signal does not end infinitely. The filter can be represented in either the time domain or frequency domain. Time-domain filter modifies time-domain input signal to time-domain output signal. The frequency domain lets us understand or to design the effect of the filter.
The discrete Fourier transformation \(\Sigma^{N-1}_{n=0} x(n)e^{-i2\pi nm/N}\) transforms time domain to frequency domain in the finite impulse response.