Ferkans — Interactive Telecom Tutor

From Multipath Sums to System Theory

Section 6.1 described multipath as a sum of delayed, scaled copies. We now formalise this: the wireless channel is a linear time-variant (LTV) filter, fully characterised by its time-variant impulse response $h(\tau; t)$ . This connects directly to the LTV framework of Section 4.5 and provides the foundation for everything that follows in this chapter.

Definition:
Time-Variant Channel Impulse Response

The time-variant impulse response (also called the input delay-spread function) is

$h(\tau; t) = \sum_{l=0}^{L-1} \alpha_l(t)\, e^{j\phi_l(t)}\, \delta(\tau - \tau_l(t))$

where $\tau$ is the delay variable and $t$ is the observation time. The received baseband signal is

$r(t) = \int_{-\infty}^{\infty} h(\tau; t)\, s(t - \tau)\, d\tau = \sum_{l=0}^{L-1} \alpha_l(t)\, e^{j\phi_l(t)}\, s(t - \tau_l(t))$

This is a convolution in $\tau$ at each time instant $t$ — the channel acts as a filter whose taps change with time.

The two-argument notation $h(\tau; t)$ uses $\tau$ as the running variable (delay) and $t$ as a parameter (time instant). This is the standard Bello convention.

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Definition:
Baseband Equivalent Channel

If the passband channel impulse response is $h_p(\tau; t)$ , the baseband equivalent channel is

$h(\tau; t) = \sum_{l=0}^{L-1} a_l(t)\, e^{-j2\pi f_0 \tau_l(t)}\, \delta(\tau - \tau_l(t))$

where $a_l(t)$ is the baseband complex amplitude. All subsequent analysis uses this baseband representation, consistent with the complex envelope formulation from Section 4.4.

Theorem: Channel Input–Output Relation

For a baseband transmitted signal $s(t)$ through a time-variant channel $h(\tau; t)$ , the received signal is

$r(t) = \int_{0}^{\tau_{\max}} h(\tau; t)\, s(t - \tau)\, d\tau + n(t)$

In the frequency domain at time $t$ :

$R(f; t) = H(f; t) \cdot S(f) + N(f)$

where $H(f; t) = \int h(\tau; t)\, e^{-j2\pi f\tau}\, d\tau$ is the time-variant transfer function.

At any frozen instant $t$ , the channel looks like an LTI filter with impulse response $h(\tau; t_0)$ . But as $t$ changes, the filter coefficients change — the channel is a snapshot-varying filter. If the channel changes slowly compared to the signal duration, we can treat it as approximately LTI over one symbol.

Proof

Derivation

The input–output relation follows directly from the superposition integral for an LTV system (Section 4.5):

$r(t) = \int h(\tau; t)\, s(t - \tau)\, d\tau$ .

Taking the Fourier transform with respect to $\tau$ (the delay variable) gives $H(f; t)$ . For a frozen $t$ , the convolution in $\tau$ becomes multiplication in $f$ :

$R(f; t) = H(f; t) \cdot S(f)$ .

Adding noise gives the complete relation. $\blacksquare$

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Why This Matters: Channel Estimation in OFDM

The time-variant transfer function $H(f; t)$ is exactly what OFDM systems estimate and equalise. Each OFDM subcarrier at frequency $f_k$ experiences a complex gain $H(f_k; t)$ . Pilot symbols are inserted at known $(f_k, t)$ locations; the receiver estimates $H(f_k; t)$ at those pilots and interpolates across the time-frequency grid. This is the foundation of modern 4G/5G channel estimation (Chapter 15).

Quick Check

If the channel impulse response is $h(\tau; t) = \alpha(t)\,\delta(\tau - \tau_0)$ (a single path with constant delay), what is the time-variant transfer function?

$H(f; t) = \alpha(t)\, e^{-j2\pi f \tau_0}$

$H(f; t) = \alpha(t)$

$H(f; t) = e^{-j2\pi f t}$

$H(f; t) = \alpha(t)\, e^{j2\pi f \tau_0}$

Correction:

H(f; t) = \alpha(t)\, e^{-j2\pi f \tau_0}

Correct. The Fourier transform of $\delta(\tau - \tau_0)$ with respect to $\tau$ is $e^{-j2\pi f \tau_0}$ , scaled by $\alpha(t)$ .

Time-Variant Impulse Response

$h(\tau; t)$ : the channel's response at time $t$ to an impulse applied $\tau$ seconds earlier. Fully characterises the LTV wireless channel.

Time-Variant Transfer Function

$H(f; t) = \mathcal{F}_\tau\{h(\tau; t)\}$ : the channel's frequency response at time $t$ . Each OFDM subcarrier experiences $H(f_k; t)$ .

The Wireless Channel as a Linear Time-Variant Filter