7.5 Application of Frequency Domain Analysis

7.5 Application of Frequency Domain Analysis

1. Stability Analysis Using Frequency Domain Methods

1.1 Nyquist Stability Criterion

1. Principle: Determines stability of closed-loop system from open-loop frequency response.

2. Nyquist Path: Contour in s-plane that encloses entire right half-plane.

3. Nyquist Plot: Plot of $G(j\omega)H(j\omega)$ as $\omega$ goes from $-\infty$ to $\infty$.

4. Stability Criterion:

   • Let $P$ = number of open-loop poles in RHP.

   • Let $N$ = net number of clockwise encirclements of point (-1, j0).

   • Closed-loop system stable if $Z = N + P = 0$, where $Z$ = closed-loop poles in RHP.

5. Gain Margin (GM):

   • Amount gain can be increased before system becomes unstable.

   • $GM = \frac{1}{|G(j\omega_p)H(j\omega_p)|}$ in linear scale.

   • $GM_{dB} = -20\log_{10}|G(j\omega_p)H(j\omega_p)|$

   • Where $\omega_p$ = phase crossover frequency (phase = -180°).

6. Phase Margin (PM):

   • Amount phase can be decreased before system becomes unstable.

   • $PM = 180° + \angle G(j\omega_g)H(j\omega_g)$

   • Where $\omega_g$ = gain crossover frequency (|GH| = 1).

7. Typical Design Specifications:

   • $GM \geq 6-10$ dB

   • $PM \geq 30-60$°

1.2 Bode Plot Analysis

1. Bode Stability Criterion:

   • System is unstable if at gain crossover frequency, phase < -180°.

   • System is stable if at gain crossover frequency, phase > -180°.

2. Relative Stability Measures:

   • Higher gain/phase margins indicate more stability.

   • Bandwidth affects system speed and noise rejection.

1.3 Root Locus Analysis in Frequency Domain

1. Relationship: Root locus shows closed-loop poles as gain varies.

2. Frequency Domain Interpretation:

   • Points on root locus correspond to specific damping ratios and natural frequencies.

   • Lines of constant damping ratio appear as radial lines from origin.

   • Lines of constant natural frequency appear as circles centered at origin.

1.4 Routh-Hurwitz Criterion Frequency Interpretation

1. Stability Boundary: Imaginary axis crossings in s-plane correspond to frequency where system becomes marginally stable.

2. Oscillation Frequency: When system is marginally stable, it oscillates at frequency given by auxiliary equation.

2. Spectral Analysis of Signals

2.1 Power Spectral Density (PSD)

1. Definition: Distribution of signal power over frequency.

2. For Continuous-Time Signals:

   • $S_{xx}(j\omega) = \lim_{T \to \infty} \frac{1}{2T} |X_T(j\omega)|^2$

   • Alternatively: $S_{xx}(j\omega) = \mathcal{F}\{R_{xx}(\tau)\}$ where $R_{xx}(\tau)$ is autocorrelation function.

3. For Discrete-Time Signals:

   • $S_{xx}(e^{j\Omega}) = \sum_{k=-\infty}^{\infty} R_{xx}[k] e^{-j\Omega k}$

4. Properties:

   • Real-valued and non-negative.

   • Even function for real signals.

   • Total power = area under PSD.

2.2 Energy Spectral Density (ESD)

1. For Energy Signals:

   • $\Psi_{xx}(j\omega) = |X(j\omega)|^2$

   • Total energy = $\frac{1}{2\pi} \int_{-\infty}^{\infty} \Psi_{xx}(j\omega) d\omega$

2.3 Estimation Methods

1. Periodogram:

   • Direct method using squared magnitude of DFT.

   • $\hat{S}_{xx}(k) = \frac{1}{N} |X[k]|^2$

2. Modified Periodogram:

   • Apply window function to reduce spectral leakage.

3. Welch's Method:

   • Divide signal into overlapping segments.

   • Average periodograms of segments.

4. Blackman-Tukey Method:

   • Estimate autocorrelation first, then Fourier transform.

2.4 Spectral Leakage and Windowing

1. Spectral Leakage: Energy spreads to adjacent frequencies due to finite observation.

2. Window Functions:

   • Rectangular: Highest resolution, most leakage.

   • Hamming: Good compromise between resolution and leakage.

   • Hanning: Better side lobe suppression.

   • Blackman: Excellent side lobe suppression, wider main lobe.

3. Window Parameters:

   • Main lobe width.

   • Side lobe level.

   • Roll-off rate.

2.5 Spectral Analysis Applications

1. Speech Processing: Formant frequency analysis.

2. Medical Signal Processing: EEG, ECG analysis.

3. Vibration Analysis: Machine fault detection.

4. Communications: Signal detection, modulation analysis.

3. Introduction to Spectrum Sensing

3.1 Basic Concepts

1. Spectrum Sensing: Detecting presence/absence of signals in frequency bands.

2. Primary Users: Licensed users with spectrum rights.

3. Secondary Users: Unlicensed users seeking spectrum opportunities.

3.2 Detection Methods

1. Energy Detection:

   • Simple, requires no prior knowledge of signal.

   • Compare received energy to threshold.

   • $T_{ED} = \sum_{n=1}^{N} |x[n]|^2$

   • Vulnerable to noise uncertainty.

2. Matched Filter Detection:

   • Optimal if signal known.

   • $T_{MF} = \sum_{n=1}^{N} x[n]s^*[n]$

   • Requires perfect synchronization.

3. Cyclostationary Feature Detection:

   • Exploit periodic statistics of modulated signals.

   • Robust to noise uncertainty.

   • Computationally intensive.

4. Covariance-Based Detection:

   • Use correlation properties of signals.

   • $T_{CD} = \frac{\frac{1}{L}\sum_{i=1}^{L} r(i)}{\sigma^2}$ where $r(i)$ = sample covariance.

3.3 Performance Metrics

1. Probability of Detection ($P_d$):

   • Correctly detecting signal when present.

2. Probability of False Alarm ($P_f$):

   • Falsely detecting signal when absent.

3. Probability of Miss ($P_m$):

   • $P_m = 1 - P_d$

4. Receiver Operating Characteristic (ROC):

   • Plot $P_d$ vs $P_f$.

   • Characterizes detector performance.

3.4 Challenges in Spectrum Sensing

1. Hidden Node Problem

2. Noise Uncertainty

3. Multipath Fading

4. Shadowing Effects

5. Computational Complexity

4. Correlation Analysis

4.1 Autocorrelation Function

1. For Continuous-Time Signals:

   • $R_{xx}(\tau) = \int_{-\infty}^{\infty} x(t)x^*(t-\tau) dt$

   • For power signals: $R_{xx}(\tau) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} x(t)x^*(t-\tau) dt$

2. For Discrete-Time Signals:

   • $R_{xx}[k] = \sum_{n=-\infty}^{\infty} x[n]x^*[n-k]$

3. Properties:

   • $R_{xx}(0) = \text{energy/power of signal}$

   • $R_{xx}(\tau) = R_{xx}^*(-\tau)$ (conjugate symmetric)

   • $|R_{xx}(\tau)| \leq R_{xx}(0)$

   • For periodic signals: $R_{xx}(\tau)$ is periodic with same period.

4.2 Cross-Correlation Function

1. For Continuous-Time Signals:

   • $R_{xy}(\tau) = \int_{-\infty}^{\infty} x(t)y^*(t-\tau) dt$

2. For Discrete-Time Signals:

   • $R_{xy}[k] = \sum_{n=-\infty}^{\infty} x[n]y^*[n-k]$

3. Properties:

   • $R_{xy}(\tau) = R_{yx}^*(-\tau)$

   • $|R_{xy}(\tau)|^2 \leq R_{xx}(0)R_{yy}(0)$

   • Not necessarily maximum at zero.

4.3 Applications of Correlation

1. Signal Detection: Match with known template.

2. Time Delay Estimation: Find peak in cross-correlation.

3. System Identification: Relate input-output correlation.

4. Pattern Recognition: Measure similarity.

5. Radar/Sonar: Range determination.

4.4 Correlation and PSD Relationship

1. Wiener-Khinchin Theorem: $S_{xx}(j\omega) = \mathcal{F}\{R_{xx}(\tau)\}$ $R_{xx}(\tau) = \mathcal{F}^{-1}\{S_{xx}(j\omega)\}$

2. Cross-Power Spectral Density: $S_{xy}(j\omega) = \mathcal{F}\{R_{xy}(\tau)\}$

4.5 Correlation Coefficients

1. Normalized Correlation: $\rho_{xy}(\tau) = \frac{R_{xy}(\tau)}{\sqrt{R_{xx}(0)R_{yy}(0)}}$

2. Range: $-1 \leq \rho_{xy}(\tau) \leq 1$

5. System Design Considerations in Frequency Domain

5.1 Filter Design Specifications

1. Frequency Response Requirements:

   • Passband ripple ($\delta_p$).

   • Stopband attenuation ($\delta_s$).

   • Transition band width.

   • Passband edge frequency ($\omega_p$).

   • Stopband edge frequency ($\omega_s$).

2. Filter Types:

   • Low-pass: Pass low frequencies, attenuate high.

   • High-pass: Pass high frequencies, attenuate low.

   • Band-pass: Pass band of frequencies.

   • Band-stop: Attenuate band of frequencies.

   • All-pass: Constant magnitude, varying phase.

5.2 Analog Filter Design

1. Butterworth Filter:

   • Maximally flat in passband.

   • $|H(j\omega)|^2 = \frac{1}{1 + (\omega/\omega_c)^{2N}}$

   • Roll-off: 20N dB/decade.

2. Chebyshev Type I:

   • Equiripple in passband, monotonic in stopband.

   • Steeper roll-off than Butterworth for same order.

3. Chebyshev Type II:

   • Monotonic in passband, equiripple in stopband.

4. Elliptic (Cauer) Filter:

   • Equiripple in both passband and stopband.

   • Sharpest transition for given order.

5. Bessel Filter:

   • Maximally flat group delay.

   • Linear phase in passband.

   • Poor frequency selectivity.

5.3 Digital Filter Design Methods

1. Impulse Invariance:

   • Sample impulse response of analog filter.

   • Preserves time-domain response.

   • May cause aliasing.

2. Bilinear Transform:

   • $s = \frac{2}{T} \frac{1-z^{-1}}{1+z^{-1}}$

   • No aliasing.

   • Warps frequency axis.

3. Frequency Sampling Method:

   • Specify desired frequency response at DFT frequencies.

   • Compute IDFT to get filter coefficients.

4. Window Method:

   • Truncate ideal impulse response with window function.

   • Trade-off between main lobe width and side lobe level.

5.4 Equalization in Frequency Domain

1. Channel Equalization:

   • Compensate for channel distortion.

   • Zero-forcing equalizer: $H_{eq}(z) = 1/H_{channel}(z)$

   • MMSE equalizer: Minimize mean square error.

2. Frequency Domain Equalization (FDE):

   • Transform to frequency domain using FFT.

   • Multiply by equalizer coefficients.

   • Transform back using IFFT.

   • Particularly effective for OFDM systems.

5.5 System Bandwidth Considerations

1. Definition: Range of frequencies where system response is within specified limits.

2. Types:

   • 3-dB Bandwidth: Frequencies where power drops to half.

   • Noise Equivalent Bandwidth: Bandwidth of ideal filter with same noise power.

   • Null-to-Null Bandwidth: Main lobe width.

3. Bandwidth-Efficiency Trade-off:

   • Higher bandwidth allows faster data rates.

   • Increased bandwidth increases noise power.

   • Regulatory constraints on spectrum usage.

5.6 Phase Response Design

1. Linear Phase Systems:

   • Constant group delay.

   • No phase distortion.

   • Symmetric impulse response.

2. Minimum Phase Systems:

   • Minimum energy delay.

   • All zeros inside unit circle.

   • Stable inverse.

3. All-Pass Systems:

   • Constant magnitude.

   • Adjust phase without affecting magnitude.

5.7 Multirate Signal Processing

1. Decimation (Downsampling):

   • Reduce sampling rate by factor M.

   • $y[n] = x[Mn]$

   • Requires anti-aliasing filter.

2. Interpolation (Upsampling):

   • Increase sampling rate by factor L.

   • Insert zeros, then apply interpolation filter.

   • $y[n] = \begin{cases} x[n/L] & n = 0, \pm L, \pm 2L, \ldots \\ 0 & \text{otherwise} \end{cases}$

3. Rational Rate Change:

   • Combine interpolation and decimation.

   • $y[n] = \text{interpolate by L, then decimate by M}$

   • Rate change factor = L/M.

6. Important Formulas and Relationships

6.1 Stability Analysis Formulas

1. Gain Margin: $GM_{dB} = -20\log_{10}|G(j\omega_p)H(j\omega_p)|$

2. Phase Margin: $PM = 180° + \angle G(j\omega_g)H(j\omega_g)$

3. Bandwidth: $\omega_{BW} = \omega_g \sqrt{\frac{1-2\zeta^2 + \sqrt{(1-2\zeta^2)^2 + 1}}{2}}$

6.2 Spectral Analysis Formulas

1. PSD from Autocorrelation: $S_{xx}(j\omega) = \mathcal{F}\{R_{xx}(\tau)\}$

2. Periodogram: $\hat{S}_{xx}(k) = \frac{1}{N} |X[k]|^2$

3. Parseval's Theorem: $\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X[k]|^2$

6.3 Correlation Formulas

1. Autocorrelation: $R_{xx}[k] = \sum_{n=-\infty}^{\infty} x[n]x^*[n-k]$

2. Cross-correlation: $R_{xy}[k] = \sum_{n=-\infty}^{\infty} x[n]y^*[n-k]$

3. Correlation Coefficient: $\rho_{xy} = \frac{R_{xy}[0]}{\sqrt{R_{xx}[0]R_{yy}[0]}}$

6.4 Filter Design Formulas

1. Bilinear Transform: $s = \frac{2}{T} \frac{1-z^{-1}}{1+z^{-1}}$

2. Butterworth Filter: $|H(j\omega)|^2 = \frac{1}{1 + (\omega/\omega_c)^{2N}}$

3. Window Function Length: $N \approx \frac{A}{(\omega_s - \omega_p)/2\pi}$ where A depends on window type.

6.5 Detection Theory Formulas

1. Signal-to-Noise Ratio: $SNR = \frac{P_{signal}}{P_{noise}}$

2. Detection Threshold: $\gamma = \sigma^2 (Q^{-1}(P_f) + \sqrt{N})$ for energy detection.

3. Probability of Detection: $P_d = Q\left(\frac{\gamma - N(\sigma_s^2 + \sigma_n^2)}{\sqrt{2N}(\sigma_s^2 + \sigma_n^2)}\right)$

6.6 System Bandwidth Relationships

1. Time-Bandwidth Product: $B \cdot T \geq \text{constant}$

   • For rectangular pulse: $B \cdot T \approx 1$

   • For Gaussian pulse: $B \cdot T = 0.5$

2. Shannon Capacity: $C = B \log_2(1 + SNR)$

3. Nyquist Rate: $R_{max} = 2B$ symbols/sec for no ISI.