Exercises

$L = 1$, $|\mathcal{A}|=2$, so the trellis has $2^1 = 2$ states.

$W_{\text{ZF}}(f) = 1/(1 + 0.6 e^{-j2\pi f})$.

$|H(f)|^2 = 1 + 0.36 + 1.2\cos(2\pi f)$ is minimized at $f=1/2$, giving $|H(1/2)|^2 = 0.16$, noise gain $1/0.16 = 6.25 \approx 8$ dB.

$W_{\text{MMSE}}(f) = \dfrac{1 + 0.6 e^{+j 2\pi f}}{1.36 + 1.2\cos(2\pi f) + 0.1}$.

At each time step, Viterbi performs $|\mathcal{A}|^L$ state updates; this repeats $T$ times, giving $O(|\mathcal{A}|^L T)$.

$\text{MMSE} = \sigma^2/(1+\sigma^2)$.

$1/\text{MMSE} - 1 = (1+\sigma^2)/\sigma^2 - 1 = 1/\sigma^2$, which is the input SNR $= \text{SINR}_{\text{out}}$ for the scalar AWGN case.

$\mathbf{f}^\star = (\mathbf{H}\mathbf{H}^H + \sigma^2\mathbf{I})^{-1} \mathbf{H}\mathbf{e}_d$, where $\mathbf{e}_d$ isolates the target delay.

$H(f) = 1 - 0.9 e^{-j2\pi f}$ has a near-null at $f=0$.

MMSE-LE amplifies noise around the null, degrading BER significantly. MLSE treats all ISI jointly and is unaffected by the null at moderate SNR, losing at most a few dB.

The whitened matched filter front end produces a minimum-phase channel impulse response with white noise output.

With correct decisions the DFE removes all post-cursor taps exactly, leaving $z[n] = x[n] + w'[n]$ with $w'$ white at the MFB noise level. The error probability matches MLSE's MFB.

$H(z) = 0.3 z^{-2}(z^2/0.3 \cdot 0.3 + 3z + 1)$ — treat as $0.3 z^2 + 0.9 z + 0.3$ with roots $z = (-0.9 \pm \sqrt{0.81-0.36})/0.6 = (-0.9\pm 0.671)/0.6$, i.e., $z \approx -0.382, -2.618$. One root is outside the unit circle, so the channel is not minimum-phase.

Reflect the outside root: replace $z = -2.618$ with its conjugate reciprocal $-1/2.618 \approx -0.382$ and rescale. This gives a minimum-phase channel with the same magnitude response.

$\log f(\mathbf{y}|\mathbf{x}) = -\|\mathbf{y}-\mathbf{h}\star\mathbf{x}\|^2 / (2\sigma^2) + \text{const}$.

Maximizing log-likelihood is minimizing the squared Euclidean distance; Viterbi accumulates this metric along paths.

The cyclic prefix makes the channel circulant; the DFT eigendecomposes circulant matrices. Per subcarrier, the signal model is scalar.

The scalar MMSE estimate is $\hat{X}_k = H_k^*/(|H_k|^2+\sigma^2) \cdot Y_k$, which is the discrete frequency sample of $W_{\text{MMSE}}(f)$.

$4^{3+1} = 256$ branch metrics per symbol, $256\cdot 100 = 25{,}600$ total.

$4^{100} \approx 1.6 \times 10^{60}$ — astronomically larger.

Transmit $x=1$ alone through the channel. The received vector is $\mathbf{r} = \mathbf{h} + \mathbf{w}$.

Matched filter output $\langle \mathbf{r}, \mathbf{h}\rangle/\|\mathbf{h}\| = \|\mathbf{h}\| + \langle\mathbf{w},\mathbf{h}\rangle/\|\mathbf{h}\|$, noise variance $\sigma^2$, signal amplitude $\|\mathbf{h}\|$, MFB = $\|\mathbf{h}\|^2/\sigma^2$.

No detector beats the MFB; MLSE and MMSE-DFE approach it, MMSE-LE can fall far below it on channels with spectral nulls.

With $\mathbf{f}^\star = (\mathbf{R}_{YY} - \mathbf{H}_{\text{post}}\mathbf{H}_{\text{post}}^H)^{-1} \mathbf{r}_{Yx}$ and $\mathbf{b}^\star = \mathbf{H}_{\text{post}}^H\mathbf{f}^\star$.

$\text{MMSE}_{\text{DFE}} = 1 - \mathbf{r}_{Yx}^H(\mathbf{R}_{YY}-\mathbf{H}_{\text{post}}\mathbf{H}_{\text{post}}^H)^{-1}\mathbf{r}_{Yx}$. Larger $L_f$ shrinks the MMSE monotonically.

$|H(f)|^2 = 1.25 - \cos(2\pi f)$; no null, but a $5$ dB dip at $f=1/2$.

MLSE is best. MMSE-DFE tracks MLSE within $\sim 0.5$ dB (no null, minimum-phase channel). MMSE-LE is $\sim 1-2$ dB worse. ZF is another $\sim 1$ dB worse than MMSE-LE and merges with MMSE only at very high SNR.