12.2 Signal processing

Signal processing is the art of extracting and manipulating information from signals. It's crucial in fields like telecommunications, audio, and biomedical engineering. This topic covers the fundamentals, from time and frequency domain analysis to sampling and quantization.

We'll explore key concepts like Fourier transforms, linear time-invariant systems, and filtering techniques. We'll also dive into spectral analysis, noise reduction, and real-world applications of signal processing in various industries.

Fundamentals of signal processing

• Signal processing involves the analysis, manipulation, and interpretation of signals to extract meaningful information or enhance signal characteristics
• Signals can be classified as continuous-time or discrete-time, and as analog or digital
• Signal processing techniques are essential in various fields, including telecommunications, audio and video processing, biomedical engineering, and radar systems

Signals in time and frequency domains

Time domain representation

Top images from around the web for Time domain representation

• 

  discrete signals - Amplitude and phase spectrum in MATLAB - Signal Processing Stack Exchange View original

  Is this image relevant?

• 

  signal processing - DSP Time domain and frequency domain - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  9. Signals, Sampling & Filtering View original

  Is this image relevant?

• 

  discrete signals - Amplitude and phase spectrum in MATLAB - Signal Processing Stack Exchange View original

  Is this image relevant?

• 

  signal processing - DSP Time domain and frequency domain - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

Top images from around the web for Time domain representation

• 

  discrete signals - Amplitude and phase spectrum in MATLAB - Signal Processing Stack Exchange View original

  Is this image relevant?

• 

  signal processing - DSP Time domain and frequency domain - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  9. Signals, Sampling & Filtering View original

  Is this image relevant?

• 

  discrete signals - Amplitude and phase spectrum in MATLAB - Signal Processing Stack Exchange View original

  Is this image relevant?

• 

  signal processing - DSP Time domain and frequency domain - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

• Represents a signal as a function of time, denoted as x(t) for continuous-time signals or x[n] for discrete-time signals
• Provides information about the signal's amplitude, duration, and temporal characteristics
• Useful for analyzing transient behavior, signal peaks, and time-based operations (time shifting, scaling)

Frequency domain representation

• Represents a signal as a function of frequency, denoted as X(f) for continuous-time signals or X(ω) for discrete-time signals
• Provides information about the signal's frequency components, magnitude, and phase
• Useful for analyzing periodic behavior, spectral content, and frequency-based operations (filtering, modulation)

Fourier transform

• Mathematical tool that converts a signal from the time domain to the frequency domain
• For continuous-time signals, the Fourier transform is defined as: $X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$
• Reveals the frequency components present in a signal and their relative amplitudes

Inverse Fourier transform

• Converts a signal from the frequency domain back to the time domain
• For continuous-time signals, the inverse Fourier transform is defined as: $x(t) = \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df$
• Allows reconstruction of the original time-domain signal from its frequency-domain representation

Linear time-invariant (LTI) systems

Impulse response

• Characterizes the response of an LTI system to a unit impulse input, denoted as h(t) for continuous-time systems or h[n] for discrete-time systems
• Completely describes the behavior of an LTI system
• Can be used to determine the output of the system for any input signal through convolution

Convolution

• Mathematical operation that combines an input signal with the impulse response of an LTI system to produce the output signal
• For continuous-time systems, the convolution integral is defined as: $y(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau$
• For discrete-time systems, the convolution sum is defined as: $y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]$

Transfer function

• Represents the frequency response of an LTI system, obtained by taking the Fourier transform of the impulse response
• For continuous-time systems, the transfer function is denoted as H(f) and is defined as: $H(f) = \int_{-\infty}^{\infty} h(t) e^{-j2\pi ft} dt$
• For discrete-time systems, the transfer function is denoted as H(z) and is obtained using the Z-transform

Stability and causality

• Stability implies that a bounded input to an LTI system produces a bounded output
• Causality requires that the output of an LTI system at any given time depends only on the current and past input values
• Stable and causal systems are essential for practical implementation and real-time signal processing applications

Sampling and quantization

Sampling theorem

• Also known as the Nyquist-Shannon sampling theorem, states that a continuous-time signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the highest frequency component in the signal
• The minimum sampling frequency required to avoid aliasing is called the Nyquist rate

Aliasing

• Occurs when a signal is sampled at a frequency lower than the Nyquist rate, causing high-frequency components to be misinterpreted as low-frequency components
• Results in distortion and loss of information in the reconstructed signal
• Can be prevented by ensuring that the sampling frequency is sufficiently high or by using an anti-aliasing filter before sampling

Quantization error

• Arises from the process of representing a continuous-amplitude signal using a finite number of discrete levels
• Introduces noise in the quantized signal, known as quantization noise
• Can be reduced by increasing the number of quantization levels or using non-uniform quantization schemes (companding)

Analog-to-digital conversion

• Process of converting a continuous-time, continuous-amplitude signal into a discrete-time, discrete-amplitude signal
• Involves sampling the analog signal at regular intervals and quantizing the sampled values
• Performed by analog-to-digital converters (ADCs) in various applications (audio recording, digital communications)

Digital-to-analog conversion

• Process of converting a discrete-time, discrete-amplitude signal back into a continuous-time, continuous-amplitude signal
• Involves reconstructing the analog signal from its digital representation using interpolation and filtering techniques
• Performed by digital-to-analog converters (DACs) in applications such as audio playback and signal generation

Discrete-time signals and systems

Discrete-time Fourier transform (DTFT)

• Extends the concept of the Fourier transform to discrete-time signals
• Represents a discrete-time signal as a function of continuous frequency, denoted as X(ω)
• Defined as: $X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}$
• Useful for analyzing the frequency content of discrete-time signals and designing discrete-time systems

Discrete Fourier transform (DFT)

• Computes the frequency-domain representation of a finite-length discrete-time signal
• Represents the signal as a sum of complex exponentials at discrete frequencies
• Defined as: $X[k] = \sum_{n=0}^{N-1} x[n] e^{-j\frac{2\pi}{N}kn}$ for k = 0, 1, ..., N-1
• Widely used in digital signal processing for spectral analysis, filtering, and data compression

Fast Fourier transform (FFT)

• Efficient algorithm for computing the DFT, reducing the computational complexity from O(N^2) to O(N log N)
• Exploits symmetries and redundancies in the DFT calculation to minimize the number of arithmetic operations
• Commonly used in real-time signal processing applications (audio, radar) where speed is critical

Z-transform

• Generalization of the discrete-time Fourier transform, extending it to complex frequencies
• Represents a discrete-time signal as a function of a complex variable z, denoted as X(z)
• Defined as: $X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$
• Useful for analyzing the stability, causality, and frequency response of discrete-time systems

Filtering techniques

Low-pass filters

• Attenuate high-frequency components of a signal while allowing low-frequency components to pass through
• Useful for removing high-frequency noise, smoothing signals, and extracting low-frequency trends
• Ideal low-pass filter has a rectangular frequency response, but practical implementations (Butterworth, Chebyshev) have a more gradual transition band

High-pass filters

• Attenuate low-frequency components of a signal while allowing high-frequency components to pass through
• Useful for removing low-frequency noise, baseline drift, and extracting high-frequency details
• Ideal high-pass filter has a rectangular frequency response, but practical implementations have a more gradual transition band

Band-pass filters

• Allow a specific range of frequencies to pass through while attenuating frequencies outside the passband
• Useful for isolating signals of interest within a specific frequency range (communication channels, biomedical signals)
• Can be designed by cascading a low-pass and a high-pass filter or using specialized filter design techniques (Butterworth, Chebyshev)

Band-stop filters

• Also known as notch filters, attenuate a specific range of frequencies while allowing frequencies outside the stopband to pass through
• Useful for removing narrow-band interference or unwanted frequency components (power line interference, acoustic feedback)
• Can be designed using specialized filter design techniques (elliptic filters) or by cascading band-pass and all-pass filters

FIR vs IIR filters

• Finite Impulse Response (FIR) filters have a finite-duration impulse response and are inherently stable and can achieve linear phase response
• Infinite Impulse Response (IIR) filters have an infinite-duration impulse response and can achieve sharper frequency selectivity with fewer coefficients, but may have stability and phase distortion issues
• Choice between FIR and IIR filters depends on the specific application requirements (stability, linear phase, computational complexity)

Spectral analysis

Power spectral density (PSD)

• Describes the distribution of power across different frequencies in a signal
• For a continuous-time signal x(t), the PSD is defined as the Fourier transform of the autocorrelation function: $S_{xx}(f) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j2\pi f\tau} d\tau$
• For a discrete-time signal x[n], the PSD is defined as the discrete-time Fourier transform of the autocorrelation sequence: $S_{xx}(\omega) = \sum_{m=-\infty}^{\infty} R_{xx}[m] e^{-j\omega m}$

Periodogram

• Estimate of the PSD based on a finite-length signal segment
• Computed by dividing the squared magnitude of the DFT of the signal segment by the segment length
• Suffers from high variance and spectral leakage due to the finite-length nature of the data
• Can be improved using windowing techniques (Hamming, Hann) to reduce spectral leakage

Welch's method

• Technique for reducing the variance of the periodogram by averaging multiple overlapping signal segments
• Divides the signal into overlapping segments, computes the periodogram for each segment, and averages the results
• Provides a trade-off between spectral resolution and variance reduction
• Commonly used in applications where a smooth and consistent PSD estimate is desired (vibration analysis, speech processing)

Spectrograms

• Time-frequency representation of a signal, showing how the spectral content of the signal changes over time
• Computed by dividing the signal into short segments, applying a window function, and calculating the PSD for each segment
• Resulting spectrogram is a 2D image with time on the x-axis, frequency on the y-axis, and color or intensity representing the signal power
• Useful for analyzing non-stationary signals and identifying time-varying frequency components (speech, music, seismic signals)

Noise reduction and signal enhancement

Signal-to-noise ratio (SNR)

• Measure of the strength of a desired signal relative to the background noise
• Defined as the ratio of the signal power to the noise power, often expressed in decibels (dB)
• Higher SNR indicates a cleaner signal with less noise contamination
• Improving the SNR is a common goal in signal processing, achieved through various noise reduction and signal enhancement techniques

Wiener filtering

• Optimal linear filter that minimizes the mean square error between the estimated signal and the desired signal
• Requires knowledge of the signal and noise power spectra or their estimates
• Performs noise reduction by attenuating frequency components with low SNR while preserving components with high SNR
• Widely used in applications such as speech enhancement, image denoising, and channel equalization

Kalman filtering

• Recursive algorithm for estimating the state of a dynamic system from noisy measurements
• Combines a system model and measurement model to predict and update the state estimate and its uncertainty
• Particularly effective for tracking and smoothing time-varying signals in the presence of noise
• Applications include navigation systems, target tracking, and biomedical signal processing

Adaptive filtering

• Class of filters that automatically adjust their parameters based on the characteristics of the input signal and the desired output
• Useful when the signal or noise characteristics are unknown or time-varying
• Common adaptive filtering algorithms include Least Mean Squares (LMS) and Recursive Least Squares (RLS)
• Applications include echo cancellation, noise cancellation, and system identification

Applications of signal processing

Audio and speech processing

• Includes tasks such as speech recognition, speaker identification, audio compression, and sound synthesis
• Techniques used include spectral analysis, linear predictive coding (LPC), and mel-frequency cepstral coefficients (MFCC)
• Applications range from virtual assistants and voice-controlled devices to audio editing and music production

Image and video processing

• Involves processing and analyzing visual data for various purposes (enhancement, compression, object recognition)
• Techniques used include spatial filtering, frequency-domain analysis (2D Fourier transform), and motion estimation
• Applications include digital photography, video surveillance, medical imaging, and computer vision

Biomedical signal processing

• Deals with the analysis and interpretation of biological signals (ECG, EEG, EMG) for diagnostic and monitoring purposes
• Techniques used include time-frequency analysis, adaptive filtering, and pattern recognition
• Applications include heart rate monitoring, brain-computer interfaces, and prosthetic control

Radar and sonar signal processing

• Involves processing signals from radar and sonar systems to detect, locate, and track targets
• Techniques used include pulse compression, Doppler processing, and beamforming
• Applications include air traffic control, maritime surveillance, and geophysical exploration