Channel Impulse Response (CIR)

The CIR is composed of a sum of discrete multipath components (MPCs) , each representing a resolvable propagation path. Each component is characterized by a complex amplitude (magnitude and phase) and an excess delay relative to the line-of-sight path. The superposition of these delayed and scaled copies of the transmitted signal causes frequency-selective fading at the receiver.

The Power Delay Profile is the squared magnitude of the CIR, representing the received power as a function of delay. Key metrics derived from the PDP include:

• Maximum Excess Delay: The largest delay at which a multipath component is detectable above the noise floor.
• RMS Delay Spread: The square root of the second central moment of the PDP, quantifying the effective time dispersion of the channel. A larger RMS delay spread indicates more severe inter-symbol interference (ISI) .

In mobile environments, the CIR is not static; it evolves over time due to relative motion between the transmitter, receiver, and scatterers. This time-selective fading is characterized by the Doppler spread, the spectral broadening of a transmitted tone. The coherence time is inversely proportional to the maximum Doppler shift and defines the interval over which the CIR remains approximately constant.

In many wideband systems, especially at mmWave frequencies, the CIR exhibits delay-domain sparsity. The number of significant multipath components is much smaller than the total number of resolvable delay bins defined by the system bandwidth. This sparsity is a fundamental property exploited by compressed sensing and deep unfolding algorithms to perform accurate channel estimation with fewer pilot symbols.

The CIR and the Channel Frequency Response (CFR) form a Fourier transform pair. While the CIR, denoted as h(τ, t), describes the channel in the delay-time domain, the CFR, denoted as H(f, t), describes the attenuation and phase shift across frequency subcarriers at a given time instant. A highly dispersive CIR corresponds to a rapidly varying CFR across frequency.

The CIR is mathematically represented as a complex-valued function. The real and imaginary components, or equivalently the in-phase (I) and quadrature (Q) components, capture both the amplitude attenuation and the carrier phase rotation induced by the propagation path. Preserving this complex structure is critical for coherent demodulation and is a key challenge for complex-valued neural networks.