通信系统知识点备忘（一）

1. Digital Modulation

1.1 Signal Space Representation

Digital modulation involves mapping digital information into analog signals suitable for transmission over physical channels. The signal space representation provides a geometric interpretation of signals, simplifying analysis and design.

• Signal Space: A vector space of functions $ x(t) $ defined over a time set $ T $.
• Inner Product:

  $$\langle x_1(t), x_2(t) \rangle = \int_{-\infty}^{\infty} x_1(t) x_2^*(t) \, dt$$

• Orthogonality: Two signals are orthogonal if their inner product is zero.
• Norm and Energy:

  $$\|x(t)\| = \sqrt{\mathcal{E}_x}, \quad \mathcal{E}_x = \int_{-\infty}^{\infty} |x(t)|^2 \, dt$$

• Distance:

  $$d(x_1, x_2) = \|x_1 - x_2\|$$

1.2 Orthonormal Bases

A set of basis functions $\{\phi_j(t)\}$ is orthonormal if:

$$\langle \phi_j, \phi_n \rangle = \delta_{jn}$$

Any signal $ s(t) $ in the subspace can be expressed as:

$$s(t) = \sum_{j=1}^N s_j \phi_j(t), \quad s_j = \langle s(t), \phi_j(t) \rangle$$

1.3 Modulation Schemes

One-Dimensional Modulation

• OOK (On-Off Keying):

  $$s_m(t) = A_m p(t), \quad A_m \in \{0, 1\}$$

• PAM (Pulse Amplitude Modulation):

  $$s_m(t) = A_m p(t), \quad A_m \in \{\pm1, \pm3, \ldots, \pm(M-1)\}$$

Two-Dimensional Modulation

• M-PSK (Phase Shift Keying):

  $$s_m(t) = g(t) \cos(2\pi f_c t + \theta_m), \quad \theta_m = \frac{2\pi m}{M}$$

• QAM (Quadrature Amplitude Modulation):

  $$s_m(t) = I(t) \cos(2\pi f_c t) - Q(t) \sin(2\pi f_c t)$$

Multi-Dimensional Modulation

• PPM (Pulse Position Modulation)
• FSK (Frequency Shift Keying)

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2. Performance Analysis

2.1 Error Probability for Binary Modulation

• Binary PAM:

  $$P_b = Q\left( \sqrt{\frac{2\mathcal{E}_b}{N_0}} \right)$$

• Binary Orthogonal Signaling:

  $$P_b = Q\left( \sqrt{\frac{\mathcal{E}_b}{N_0}} \right)$$

Binary PAM outperforms orthogonal signaling for the same energy.

2.2 M-ary Modulation

M-PAM

• Symbol Error Probability:

  $$P_e = \frac{2(M-1)}{M} Q\left( \sqrt{\frac{6(\log_2 M) \mathcal{E}_b}{(M^2-1)N_0}} \right)$$

M-PSK

• For QPSK:

  $$P_e = 2P_{\text{BPSK}} - P_{\text{BPSK}}^2$$

• With Gray coding:

  $$P_b \approx \frac{P_e}{\log_2 M}$$

M-QAM

• Rectangular QAM:

  $$P_e \approx 4\left(1 - \frac{1}{\sqrt{M}}\right) Q\left( \sqrt{\frac{3\log_2 M}{M-1} \cdot \frac{\mathcal{E}_b}{N_0}} \right)$$

M-ary Orthogonal Signaling

• Symbol Error Probability:

  $$P_e \leq (M-1) Q\left( \sqrt{\frac{\mathcal{E}_s}{N_0}} \right)$$

• Bit Error Probability:

  $$P_b \approx \frac{P_e}{2}$$

2.3 Spectral Efficiency

• Spectral Efficiency:

  $$\nu = \frac{R_b}{W} \quad \text{(bits/s/Hz)}$$

• PAM, PSK, QAM: $ \nu \to \infty $ as $ M \to \infty $
• Orthogonal signaling (PPM, FSK): $ \nu \to 0 $ as $ M \to \infty $

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3. Optimal Receiver Design

3.1 AWGN Channel Model

Received signal:

$$r(t) = s_m(t) + n(t), \quad n(t) \sim \mathcal{N}(0, N_0/2)$$

3.2 Demodulation

Correlation-Type Demodulator

$$r_k = \int_{0}^{T} r(t) \phi_k(t) \, dt = s_{mk} + n_k$$

Matched Filter Demodulator

$$h_k(t) = \phi_k(T - t), \quad r_k = (r \star h_k)(T)$$

3.3 Detection

MAP Detector

$$\hat{m} = \arg\max_{m} P(s_m | \mathbf{r}) = \arg\max_{m} p(\mathbf{r}|s_m) P(s_m)$$

ML Detector (for equiprobable symbols)

$$\hat{m} = \arg\max_{m} p(\mathbf{r}|s_m) = \arg\min_{m} \|\mathbf{r} - \mathbf{s}_m\|$$

3.4 Noncoherent and Differential Demodulation

• Noncoherent OOK/FSK: Envelope detection to handle phase mismatch.
• Differential Modulation (DBPSK/DQPSK): Encodes information in phase changes to combat fixed phase offsets.