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)

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 $

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.

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