The document proposes methods for primary user detection in cognitive radio that are robust to noise uncertainty (NU). It derives closed-form probability density functions of the signal and energy statistics under NU, allowing an optimal detector to be employed. It models NU using a log-normal distribution and evaluates detection performance for the energy detector. The proposed detector that accounts for the NU distribution can avoid the "SNR wall" phenomenon and achieve better detection performance than a worst-case detector that is not informed by the NU distribution.
Noise uncertainty in cognitive radio sensing analytical modeling and detectio...Marwan Hammouda
This document summarizes a paper that analyzes noise uncertainty in cognitive radio signal detection. It proposes modeling the noise process statistically when there is noise uncertainty present. Specifically, it models the inverse noise standard deviation with a Gaussian distribution and shows it agrees well with the more common lognormal distribution for low to moderate noise uncertainty. It derives closed-form probability density functions for noise samples and energy of multiple samples, allowing optimal detection even with noise uncertainty. Initial measurements explore energy detection at low SNR, demonstrating noise calibration can provide useful detection down to -16 dB and noise uncertainty is not significant for instrument-grade low-noise amplifiers over sub-minute acquisition times.
This document discusses energy detection of unknown signals in fading environments. It proposes modeling the received signal power distribution under combined slow and fast fading. This allows deriving the distribution of the detector's decision variable in closed form. Specifically:
1) It models the received signal as the sum of the signal and noise, scaled by a complex channel amplitude representing fast and slow fading.
2) It derives an expression for the sufficient statistic at the detector's output and simplifies it under assumptions of high sample numbers and independent samples.
3) It expresses the distribution of the decision variable as an integral of the distribution for a fixed SNR, averaged over the SNR distribution due to fading.
4) It provides the specific
Explicit Signal to Noise Ratio in Reproducing Kernel Hilbert Spaces.pdfgrssieee
This document presents a new nonlinear kernel feature extraction method called Kernel Minimum Noise Fraction (KMNF) for remote sensing data. KMNF is based on the Minimum Noise Fraction transformation but estimates noise explicitly in a reproducing kernel Hilbert space, allowing it to handle nonlinear relationships between signal and noise features. The authors introduce KMNF and compare it to other feature extraction methods like PCA, MNF, and KPCA on a hyperspectral image classification task.
Zero-padding a signal involves appending artificial zeros to increase the length of the signal. This increases the frequency resolution of the discrete Fourier transform (DFT) by changing the implicit periodicity assumption made about the signal. Specifically, zero-padding moves the DFT closer to approximating the true discrete-time Fourier transform (DTFT) by changing the assumption from periodicity to assuming the signal is zero outside the observed range. While zero-padding does not provide new information, it can help reveal features of a signal by modifying the implicit assumptions of the DFT.
The document describes an experiment on linear time invariant systems. The objectives are to: 1) Convolve a signal with an impulse response, 2) Find step responses using the impulse response for rectangular, exponential and sinusoidal inputs, 3) Show stable and unstable conditions using pole-zero plots, 4) Apply filtering to an image using circular convolution with overlap add and save methods. Background topics discussed are aliasing, impulse/step inputs, and even/odd signals. Questions involve plotting a signal, its Fourier transform, and filtering an image.
DSP_2018_FOEHU - Lec 08 - The Discrete Fourier TransformAmr E. Mohamed
The document provides an overview of the Discrete Fourier Transform (DFT). It begins by discussing limitations of the discrete-time Fourier transform (DTFT) and z-transform in that they are defined for infinite sequences and continuous variables. The DFT avoids these issues by being a numerically computable transform for finite discrete-time signals. It works by taking a finite signal, making it periodic, and computing its discrete Fourier transform which is a discrete frequency spectrum. This makes the DFT highly suitable for digital signal processing. The document then provides details on computation of the DFT and its relationship to the DTFT and z-transform.
DSP_FOEHU - Lec 03 - Sampling of Continuous Time SignalsAmr E. Mohamed
1. The Nyquist interval is the longest time interval that can be used for sampling a bandlimited signal while still allowing reconstruction of the signal without distortion.
2. The sampling theorem states that a signal x(t) with finite energy can be reconstructed from its sampled values x(nTs) if the sampling frequency is greater than twice the maximum frequency of the signal.
3. Reconstruction of a sampled signal involves representing the sampled signal as a sum of sinusoids with frequencies that are integer multiples of the sampling frequency below the Nyquist frequency.
1) Sparse signal processing techniques aim to represent signals using a small number of nonzero coefficients.
2) Compressive sensing (CS) allows acquiring signals at a rate below Nyquist by taking linear measurements using an incoherent sensing matrix.
3) CS reconstruction recovers the original sparse signal by imposing sparsity constraints during recovery from the undersampled measurements. The number of measurements required depends on the sparsity and mutual incoherence between the sensing and sparsity bases.
Noise uncertainty in cognitive radio sensing analytical modeling and detectio...Marwan Hammouda
This document summarizes a paper that analyzes noise uncertainty in cognitive radio signal detection. It proposes modeling the noise process statistically when there is noise uncertainty present. Specifically, it models the inverse noise standard deviation with a Gaussian distribution and shows it agrees well with the more common lognormal distribution for low to moderate noise uncertainty. It derives closed-form probability density functions for noise samples and energy of multiple samples, allowing optimal detection even with noise uncertainty. Initial measurements explore energy detection at low SNR, demonstrating noise calibration can provide useful detection down to -16 dB and noise uncertainty is not significant for instrument-grade low-noise amplifiers over sub-minute acquisition times.
This document discusses energy detection of unknown signals in fading environments. It proposes modeling the received signal power distribution under combined slow and fast fading. This allows deriving the distribution of the detector's decision variable in closed form. Specifically:
1) It models the received signal as the sum of the signal and noise, scaled by a complex channel amplitude representing fast and slow fading.
2) It derives an expression for the sufficient statistic at the detector's output and simplifies it under assumptions of high sample numbers and independent samples.
3) It expresses the distribution of the decision variable as an integral of the distribution for a fixed SNR, averaged over the SNR distribution due to fading.
4) It provides the specific
Explicit Signal to Noise Ratio in Reproducing Kernel Hilbert Spaces.pdfgrssieee
This document presents a new nonlinear kernel feature extraction method called Kernel Minimum Noise Fraction (KMNF) for remote sensing data. KMNF is based on the Minimum Noise Fraction transformation but estimates noise explicitly in a reproducing kernel Hilbert space, allowing it to handle nonlinear relationships between signal and noise features. The authors introduce KMNF and compare it to other feature extraction methods like PCA, MNF, and KPCA on a hyperspectral image classification task.
Zero-padding a signal involves appending artificial zeros to increase the length of the signal. This increases the frequency resolution of the discrete Fourier transform (DFT) by changing the implicit periodicity assumption made about the signal. Specifically, zero-padding moves the DFT closer to approximating the true discrete-time Fourier transform (DTFT) by changing the assumption from periodicity to assuming the signal is zero outside the observed range. While zero-padding does not provide new information, it can help reveal features of a signal by modifying the implicit assumptions of the DFT.
The document describes an experiment on linear time invariant systems. The objectives are to: 1) Convolve a signal with an impulse response, 2) Find step responses using the impulse response for rectangular, exponential and sinusoidal inputs, 3) Show stable and unstable conditions using pole-zero plots, 4) Apply filtering to an image using circular convolution with overlap add and save methods. Background topics discussed are aliasing, impulse/step inputs, and even/odd signals. Questions involve plotting a signal, its Fourier transform, and filtering an image.
DSP_2018_FOEHU - Lec 08 - The Discrete Fourier TransformAmr E. Mohamed
The document provides an overview of the Discrete Fourier Transform (DFT). It begins by discussing limitations of the discrete-time Fourier transform (DTFT) and z-transform in that they are defined for infinite sequences and continuous variables. The DFT avoids these issues by being a numerically computable transform for finite discrete-time signals. It works by taking a finite signal, making it periodic, and computing its discrete Fourier transform which is a discrete frequency spectrum. This makes the DFT highly suitable for digital signal processing. The document then provides details on computation of the DFT and its relationship to the DTFT and z-transform.
DSP_FOEHU - Lec 03 - Sampling of Continuous Time SignalsAmr E. Mohamed
1. The Nyquist interval is the longest time interval that can be used for sampling a bandlimited signal while still allowing reconstruction of the signal without distortion.
2. The sampling theorem states that a signal x(t) with finite energy can be reconstructed from its sampled values x(nTs) if the sampling frequency is greater than twice the maximum frequency of the signal.
3. Reconstruction of a sampled signal involves representing the sampled signal as a sum of sinusoids with frequencies that are integer multiples of the sampling frequency below the Nyquist frequency.
1) Sparse signal processing techniques aim to represent signals using a small number of nonzero coefficients.
2) Compressive sensing (CS) allows acquiring signals at a rate below Nyquist by taking linear measurements using an incoherent sensing matrix.
3) CS reconstruction recovers the original sparse signal by imposing sparsity constraints during recovery from the undersampled measurements. The number of measurements required depends on the sparsity and mutual incoherence between the sensing and sparsity bases.
Nyquist criterion for distortion less baseband binary channelPriyangaKR1
binary transmission system
From design point of view – frequency response of the channel and transmitted pulse shape are specified; the frequency response of the transmit and receive filters has to be determined so as to reconstruct [bk]
This document discusses correlative-level coding and its applications in baseband pulse transmission systems. Correlative-level coding introduces controlled intersymbol interference to increase signaling rate. It allows partial response signaling and maximum likelihood detection at the receiver. Specific techniques discussed include duobinary signaling and modified duobinary signaling. The document also covers tapped-delay line equalization using adaptive algorithms like least mean square to compensate for channel distortion. Decision feedback equalization and its implementation are summarized as well. Eye patterns are described as a tool to evaluate signal quality in such systems.
This document presents a new DFT-based approach for detecting and correcting gain mismatch in time-interleaved ADCs. It introduces the gain mismatch problem in TI-ADCs and how it reduces the spurious free dynamic range. The proposed method uses the discrete Fourier transform to detect the gain mismatch between ADC sub-channels based on the difference between the ideal DFT and actual DFT. It then introduces a feedback system using this difference signal to iteratively correct the gain mismatch. Simulation results show the approach improves SFDR by more than 30dB by correcting a ±2% gain mismatch in a two-channel TI-ADC.
- Compressive sensing (CS) theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use
- CS relies on two principle :
sparsity: which pertains to the signal of interest
In coherence : which pertains to the sensing modality
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
DSP_2018_FOEHU - Lec 02 - Sampling of Continuous Time SignalsAmr E. Mohamed
The document discusses sampling of continuous-time signals. It defines different types of signals and sampling methods. Ideal sampling involves multiplying the signal by a train of impulse functions to select sample values at regular intervals. For practical sampling, a train of rectangular pulses is used to approximate ideal sampling. Flat-top sampling is achieved by convolving the ideally sampled signal with a rectangular pulse, resulting in samples held at a constant height for the sample period. The Nyquist sampling theorem states that a signal must be sampled at least twice its maximum frequency to avoid aliasing when reconstructing the original signal from samples. An anti-aliasing filter can be used before sampling to prevent aliasing from high frequencies above half the sampling rate.
An Optimized Transform for ECG Signal CompressionIDES Editor
A significant feature of the coming digital era is the
exponential increase in digital data, obtained from various
signals specially the biomedical signals such as
electrocardiogram (ECG), electroencephalogram (EEG),
electromyogram (EMG) etc. How to transmit or store these
signals efficiently becomes the most important issue. A digital
compression technique is often used to solve this problem.
This paper proposed a comparative study of transform based
approach for ECG signal compression. Adaptive threshold is
used on the transformed coefficients. The algorithm is tested
for 10 different records from MIT-BIH arrhythmia database
and obtained percentage root mean difference as around
0.528 to 0.584% for compression ratio of 18.963:1 to 23.011:1
for DWT. Among DFT, DCT and DWT techniques, DWT has
been proven to be very efficient for ECG signal coding.
Further improvement in the CR is possible by efficient
entropy coding.
Dsp 2018 foehu - lec 10 - multi-rate digital signal processingAmr E. Mohamed
This document discusses multi-rate digital signal processing and concepts related to sampling continuous-time signals. It begins by introducing discrete-time processing of continuous signals using an ideal continuous-to-discrete converter. It then covers the Nyquist sampling theorem and relationships between continuous and discrete Fourier transforms. It discusses ideal and practical reconstruction using zero-order hold and anti-imaging filters. Finally, it introduces the concepts of downsampling and upsampling in multi-rate digital signal processing systems.
The document discusses the discrete Fourier transform (DFT) and its applications. It provides an overview of DFT and how it represents a signal in the frequency domain. It then describes the fast Fourier transform (FFT) algorithm, which efficiently computes the DFT. The document outlines algorithms to compute the inverse DFT and circular convolution using the DFT. It includes MATLAB code implementations of DFT, inverse DFT, FFT, and circular convolution. Graphs are shown comparing computation times of the algorithms.
A Novel Methodology for Designing Linear Phase IIR FiltersIDES Editor
This paper presents a novel technique for
designing an Infinite Impulse Response (IIR) Filter with
Linear Phase Response. The design of IIR filter is always a
challenging task due to the reason that a Linear Phase
Response is not realizable in this kind. The conventional
techniques involve large number of samples and higher
order filter for better approximation resulting in complex
hardware for implementing the same. In addition, an
extensive computational resource for obtaining the inverse
of huge matrices is required. However, we propose a
technique, which uses the frequency domain sampling along
with the linear programming concept to achieve a filter
design, which gives a best approximation for the linear
phase response. The proposed method can give the closest
response with less number of samples (only 10) and is
computationally simple. We have presented the filter design
along with its formulation and solving methodology.
Numerical results are used to substantiate the efficiency of
the proposed method.
DSP_2018_FOEHU - Lec 06 - FIR Filter DesignAmr E. Mohamed
This lecture discusses the design of finite impulse response (FIR) filters. It introduces the window method for FIR filter design, which involves truncating the ideal impulse response with a window function to obtain a causal FIR filter. Common window functions are presented such as rectangular, triangular, Hanning, Hamming, and Blackman windows. These windows trade off main lobe width and side lobe levels. The document provides an example design of a low-pass FIR filter using the Hamming window to meet given passband and stopband specifications.
The document discusses the Fast Fourier Transform (FFT) algorithm.
1) The FFT is a set of techniques that exploits symmetries in the Discrete Fourier Transform (DFT) to make its computation much faster. The speedup increases with larger DFT sizes.
2) The Cooley-Tukey algorithm decomposes an N-point DFT into smaller DFTs by splitting the indices, resulting in an algorithm that is proportional to NlogN operations rather than N^2.
3) The algorithm can be represented as a series of "butterfly" operations, with each butterfly requiring only 2 multiplications. This reduces the number of multiplications needed compared to direct computation of the DFT.
1) The document discusses different types of small-scale fading that can occur in radio propagation, including Rayleigh fading and Rician fading.
2) Rayleigh fading results when there is no line-of-sight path between transmitter and receiver. It follows a Rayleigh distribution.
3) Rician fading results when there is a dominant line-of-sight path in addition to other scattered paths. It follows a Rician distribution characterized by a Rician factor K.
1) The document describes digital signal detection techniques at the receiver of a digital communication system.
2) It discusses the maximum a posteriori probability (MAP) and maximum likelihood (ML) detection criteria. The ML criterion reduces to choosing the signal that minimizes the Euclidean distance between the received signal vector and possible transmitted signals.
3) Detection errors occur when the received signal, distorted by noise, falls inside the decision region of another signal. The probability of error depends on the noise distribution around the actual transmitted signal.
Digital Signal Processing[ECEG-3171]-Ch1_L06Rediet Moges
This document summarizes key concepts about analog reconstruction and changing sampling rates from a lecture on sampling and reconstruction:
1) Analog reconstruction involves converting samples to impulse trains, then filtering with an ideal low-pass reconstruction filter to perfectly reconstruct the original signal. A staircase reconstructor is commonly used instead as it is realizable but does not eliminate replicas completely.
2) Downsampling reduces the sampling rate by an integer M by keeping every Mth sample. Upsampling increases the rate by inserting zeros between samples. Both can be done without aliasing by first low-pass filtering at the Nyquist rate of the new sampling frequency.
3) Changing the sampling rate by a rational factor can be achieved by cascading
Dsp U Lec07 Realization Of Discrete Time Systemstaha25
This document provides an overview of discrete-time systems and digital signal processing. It discusses discrete-time system components like unit delays and adders. It also covers discrete system networks including FIR and IIR networks. Various realizations of discrete systems are presented, including direct form I and II, cascaded, and parallel realizations. Digital filters are defined and the advantages and disadvantages as well as types (FIR and IIR) are discussed. Design steps and specifications for digital filters are also outlined.
This document contains questions and answers related to digital signal processing. It discusses key concepts such as signals, systems, analog and digital signals, discrete time signals, digital signal processing, advantages of DSP, applications of DSP, discrete time systems, obtaining discrete time signals from continuous time signals, impulse response and its significance, discrete convolution, importance of linear convolution in DSP, circular convolution, periodic convolution, importance of circular convolution in DSP, performing linear convolution using circular convolution, correlation, auto-correlation, differences between discrete time Fourier transform and discrete Fourier transform, advantages of using discrete Fourier transform in computers, periodic convolution, need for fast Fourier transform, definition of fast Fourier transform, differences between DIT and DIF fast Fourier
This document discusses a proposed architecture for a higher Nyquist-range digital-to-analog converter (DAC) that employs sinusoidal interpolation.
[1] Conventional DACs operate within the Nyquist range, but the proposed architecture aims to utilize higher Nyquist ranges by approximating an oscillating signal from an RF DAC concept using sinusoidal interpolation in the time domain.
[2] The proposed architecture quantizes both the input signal and pulse amplitude modulation waveform and combines them digitally, replacing analog oscillatory circuits with a digital data stream. This reduces analog complexity compared to existing techniques.
[3] Simulation results and theoretical analysis are presented to support that the proposed architecture can provide similar performance
This document analyzes the performance of energy detection algorithms for spectrum sensing in cognitive radio systems. It discusses how energy detection works by formulating the spectrum sensing problem as a binary hypothesis test to determine if a primary user is present or absent. It finds that increasing the signal-to-noise ratio, sample size, or dynamic detection threshold can improve detection performance. However, it also notes that energy detection is very sensitive to noise uncertainty, which can seriously degrade performance, especially in low signal-to-noise environments. A dynamic thresholding approach is proposed to improve robustness to noise uncertainty.
The document summarizes Lecture 7 which covered:
1) A review of Lecture 6 on PCM waveforms and the remaining portion of Chapter 2 on spectral densities of PCM waveforms and multi-level signaling.
2) An overview of Chapter 3 on baseband demodulation/detection including matched filters, correlators, Bayes' decision criterion, and maximum likelihood detection.
3) Key aspects of line codes including how pulse shaping can control the signal spectrum and ensure symbol transitions, comparisons of line codes based on power spectral density, DC component, and bandwidth.
Nyquist criterion for distortion less baseband binary channelPriyangaKR1
binary transmission system
From design point of view – frequency response of the channel and transmitted pulse shape are specified; the frequency response of the transmit and receive filters has to be determined so as to reconstruct [bk]
This document discusses correlative-level coding and its applications in baseband pulse transmission systems. Correlative-level coding introduces controlled intersymbol interference to increase signaling rate. It allows partial response signaling and maximum likelihood detection at the receiver. Specific techniques discussed include duobinary signaling and modified duobinary signaling. The document also covers tapped-delay line equalization using adaptive algorithms like least mean square to compensate for channel distortion. Decision feedback equalization and its implementation are summarized as well. Eye patterns are described as a tool to evaluate signal quality in such systems.
This document presents a new DFT-based approach for detecting and correcting gain mismatch in time-interleaved ADCs. It introduces the gain mismatch problem in TI-ADCs and how it reduces the spurious free dynamic range. The proposed method uses the discrete Fourier transform to detect the gain mismatch between ADC sub-channels based on the difference between the ideal DFT and actual DFT. It then introduces a feedback system using this difference signal to iteratively correct the gain mismatch. Simulation results show the approach improves SFDR by more than 30dB by correcting a ±2% gain mismatch in a two-channel TI-ADC.
- Compressive sensing (CS) theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use
- CS relies on two principle :
sparsity: which pertains to the signal of interest
In coherence : which pertains to the sensing modality
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
DSP_2018_FOEHU - Lec 02 - Sampling of Continuous Time SignalsAmr E. Mohamed
The document discusses sampling of continuous-time signals. It defines different types of signals and sampling methods. Ideal sampling involves multiplying the signal by a train of impulse functions to select sample values at regular intervals. For practical sampling, a train of rectangular pulses is used to approximate ideal sampling. Flat-top sampling is achieved by convolving the ideally sampled signal with a rectangular pulse, resulting in samples held at a constant height for the sample period. The Nyquist sampling theorem states that a signal must be sampled at least twice its maximum frequency to avoid aliasing when reconstructing the original signal from samples. An anti-aliasing filter can be used before sampling to prevent aliasing from high frequencies above half the sampling rate.
An Optimized Transform for ECG Signal CompressionIDES Editor
A significant feature of the coming digital era is the
exponential increase in digital data, obtained from various
signals specially the biomedical signals such as
electrocardiogram (ECG), electroencephalogram (EEG),
electromyogram (EMG) etc. How to transmit or store these
signals efficiently becomes the most important issue. A digital
compression technique is often used to solve this problem.
This paper proposed a comparative study of transform based
approach for ECG signal compression. Adaptive threshold is
used on the transformed coefficients. The algorithm is tested
for 10 different records from MIT-BIH arrhythmia database
and obtained percentage root mean difference as around
0.528 to 0.584% for compression ratio of 18.963:1 to 23.011:1
for DWT. Among DFT, DCT and DWT techniques, DWT has
been proven to be very efficient for ECG signal coding.
Further improvement in the CR is possible by efficient
entropy coding.
Dsp 2018 foehu - lec 10 - multi-rate digital signal processingAmr E. Mohamed
This document discusses multi-rate digital signal processing and concepts related to sampling continuous-time signals. It begins by introducing discrete-time processing of continuous signals using an ideal continuous-to-discrete converter. It then covers the Nyquist sampling theorem and relationships between continuous and discrete Fourier transforms. It discusses ideal and practical reconstruction using zero-order hold and anti-imaging filters. Finally, it introduces the concepts of downsampling and upsampling in multi-rate digital signal processing systems.
The document discusses the discrete Fourier transform (DFT) and its applications. It provides an overview of DFT and how it represents a signal in the frequency domain. It then describes the fast Fourier transform (FFT) algorithm, which efficiently computes the DFT. The document outlines algorithms to compute the inverse DFT and circular convolution using the DFT. It includes MATLAB code implementations of DFT, inverse DFT, FFT, and circular convolution. Graphs are shown comparing computation times of the algorithms.
A Novel Methodology for Designing Linear Phase IIR FiltersIDES Editor
This paper presents a novel technique for
designing an Infinite Impulse Response (IIR) Filter with
Linear Phase Response. The design of IIR filter is always a
challenging task due to the reason that a Linear Phase
Response is not realizable in this kind. The conventional
techniques involve large number of samples and higher
order filter for better approximation resulting in complex
hardware for implementing the same. In addition, an
extensive computational resource for obtaining the inverse
of huge matrices is required. However, we propose a
technique, which uses the frequency domain sampling along
with the linear programming concept to achieve a filter
design, which gives a best approximation for the linear
phase response. The proposed method can give the closest
response with less number of samples (only 10) and is
computationally simple. We have presented the filter design
along with its formulation and solving methodology.
Numerical results are used to substantiate the efficiency of
the proposed method.
DSP_2018_FOEHU - Lec 06 - FIR Filter DesignAmr E. Mohamed
This lecture discusses the design of finite impulse response (FIR) filters. It introduces the window method for FIR filter design, which involves truncating the ideal impulse response with a window function to obtain a causal FIR filter. Common window functions are presented such as rectangular, triangular, Hanning, Hamming, and Blackman windows. These windows trade off main lobe width and side lobe levels. The document provides an example design of a low-pass FIR filter using the Hamming window to meet given passband and stopband specifications.
The document discusses the Fast Fourier Transform (FFT) algorithm.
1) The FFT is a set of techniques that exploits symmetries in the Discrete Fourier Transform (DFT) to make its computation much faster. The speedup increases with larger DFT sizes.
2) The Cooley-Tukey algorithm decomposes an N-point DFT into smaller DFTs by splitting the indices, resulting in an algorithm that is proportional to NlogN operations rather than N^2.
3) The algorithm can be represented as a series of "butterfly" operations, with each butterfly requiring only 2 multiplications. This reduces the number of multiplications needed compared to direct computation of the DFT.
1) The document discusses different types of small-scale fading that can occur in radio propagation, including Rayleigh fading and Rician fading.
2) Rayleigh fading results when there is no line-of-sight path between transmitter and receiver. It follows a Rayleigh distribution.
3) Rician fading results when there is a dominant line-of-sight path in addition to other scattered paths. It follows a Rician distribution characterized by a Rician factor K.
1) The document describes digital signal detection techniques at the receiver of a digital communication system.
2) It discusses the maximum a posteriori probability (MAP) and maximum likelihood (ML) detection criteria. The ML criterion reduces to choosing the signal that minimizes the Euclidean distance between the received signal vector and possible transmitted signals.
3) Detection errors occur when the received signal, distorted by noise, falls inside the decision region of another signal. The probability of error depends on the noise distribution around the actual transmitted signal.
Digital Signal Processing[ECEG-3171]-Ch1_L06Rediet Moges
This document summarizes key concepts about analog reconstruction and changing sampling rates from a lecture on sampling and reconstruction:
1) Analog reconstruction involves converting samples to impulse trains, then filtering with an ideal low-pass reconstruction filter to perfectly reconstruct the original signal. A staircase reconstructor is commonly used instead as it is realizable but does not eliminate replicas completely.
2) Downsampling reduces the sampling rate by an integer M by keeping every Mth sample. Upsampling increases the rate by inserting zeros between samples. Both can be done without aliasing by first low-pass filtering at the Nyquist rate of the new sampling frequency.
3) Changing the sampling rate by a rational factor can be achieved by cascading
Dsp U Lec07 Realization Of Discrete Time Systemstaha25
This document provides an overview of discrete-time systems and digital signal processing. It discusses discrete-time system components like unit delays and adders. It also covers discrete system networks including FIR and IIR networks. Various realizations of discrete systems are presented, including direct form I and II, cascaded, and parallel realizations. Digital filters are defined and the advantages and disadvantages as well as types (FIR and IIR) are discussed. Design steps and specifications for digital filters are also outlined.
This document contains questions and answers related to digital signal processing. It discusses key concepts such as signals, systems, analog and digital signals, discrete time signals, digital signal processing, advantages of DSP, applications of DSP, discrete time systems, obtaining discrete time signals from continuous time signals, impulse response and its significance, discrete convolution, importance of linear convolution in DSP, circular convolution, periodic convolution, importance of circular convolution in DSP, performing linear convolution using circular convolution, correlation, auto-correlation, differences between discrete time Fourier transform and discrete Fourier transform, advantages of using discrete Fourier transform in computers, periodic convolution, need for fast Fourier transform, definition of fast Fourier transform, differences between DIT and DIF fast Fourier
This document discusses a proposed architecture for a higher Nyquist-range digital-to-analog converter (DAC) that employs sinusoidal interpolation.
[1] Conventional DACs operate within the Nyquist range, but the proposed architecture aims to utilize higher Nyquist ranges by approximating an oscillating signal from an RF DAC concept using sinusoidal interpolation in the time domain.
[2] The proposed architecture quantizes both the input signal and pulse amplitude modulation waveform and combines them digitally, replacing analog oscillatory circuits with a digital data stream. This reduces analog complexity compared to existing techniques.
[3] Simulation results and theoretical analysis are presented to support that the proposed architecture can provide similar performance
This document analyzes the performance of energy detection algorithms for spectrum sensing in cognitive radio systems. It discusses how energy detection works by formulating the spectrum sensing problem as a binary hypothesis test to determine if a primary user is present or absent. It finds that increasing the signal-to-noise ratio, sample size, or dynamic detection threshold can improve detection performance. However, it also notes that energy detection is very sensitive to noise uncertainty, which can seriously degrade performance, especially in low signal-to-noise environments. A dynamic thresholding approach is proposed to improve robustness to noise uncertainty.
The document summarizes Lecture 7 which covered:
1) A review of Lecture 6 on PCM waveforms and the remaining portion of Chapter 2 on spectral densities of PCM waveforms and multi-level signaling.
2) An overview of Chapter 3 on baseband demodulation/detection including matched filters, correlators, Bayes' decision criterion, and maximum likelihood detection.
3) Key aspects of line codes including how pulse shaping can control the signal spectrum and ensure symbol transitions, comparisons of line codes based on power spectral density, DC component, and bandwidth.
Wave-packet Treatment of Neutrinos and Its Quantum-mechanical ImplicationsCheng-Hsien Li
The document discusses the wave-packet treatment of neutrinos and its implications. It defines the volume occupied by a neutrino wave packet based on its probability distribution. It then introduces the concept of overlap factor to quantify how likely neutrino wave packets from a source overlap in the detector. The overlap factor depends on source intensity, neutrino energy, and geometric factors. It is estimated that the overlap could be significant for neutrinos from radioactive sources but negligible for accelerator and reactor neutrinos. For astrophysical sources like the Sun and supernovae, the overlap is expected to be overwhelming given their intense fluxes.
Speech signal time frequency representationNikolay Karpov
This lecture discusses spectrogram analysis and the short-term discrete Fourier transform. It defines normalized time and frequency, examines the effect of window length on time-frequency resolution, and derives descriptions of frequency and time resolution. It also reviews properties of the discrete Fourier transform and illustrates the uncertainty principle with examples.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
The document discusses online identification and tracking of signal subspaces from highly incomplete information. It proposes using an iterative natural gradient descent approach on the Grassmannian manifold to update the subspace matrix based on new observations. Specifically, it uses the singular value decomposition of the natural gradient and an updating rule to perform gradient descent in the Grassmannian. Simulation results demonstrate the effectiveness of this Grassmannian Optimization Under Subspace Tracking (GROUSE) algorithm.
Bayesian adaptive optimal estimation using a sieve priorJulyan Arbel
This document presents results on Bayesian optimal adaptive estimation using a sieve prior. It derives posterior concentration rates and risk convergence rates for models that accommodate a sieve prior. For the Gaussian white noise model, it shows the rates are adaptive optimal under global loss but a lower bound on the rate is obtained under pointwise loss, indicating the sieve prior is not optimal. Further work on posterior concentration rates under pointwise loss is suggested.
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Noise Uncertainty in Cognitive Radio Analytical Modeling and Detection Performance
1. Noise Uncertainty in Cognitive Radio
Analytical Modeling and Detection Performance
Marwan A. Hammouda
Supervisor: Prof. Jon Wallace
Jacobs University Bremen
June 19, 2012
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
2. Outlines
Motivation
Introduction
Cognitive Radio
Primary Sensing
Noise Uncertainty NU
System Model
General Assumptions
Noise Uncertainty Model.
Detection with NU
Case 1: Uncorrelated Signals
Case 1: Correlated Signals
Noise Calibration Measurments
Conclusion
Future Works
Published Work
References
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
3. Motivation
Methods for primary user detection in cognitive radio may be severely
impaired by noise uncertainty (NU) and the associated SNR wall
phenomenon.
Propose the ability to avoid the SNR wall by detailed statistical modeling
of the noise process when NU is present.
Derive closed-form pdfs of signal and energy under NU, allowing an
optimal Neyman-Pearson detector to be employed when NU is present.
Explore energy detector at low SNR in a practical system.
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
4. Introduction
Cognitive Radio
Cognitive Radio is an interesting emerging paradigm for radio networks.
Basically aims at improving the spectrum utilization where radios can
sense and exploit unused spectrum
Allow networks to operate in a more decentralized fashion.
Challenge: Require low missed detection at low SNR
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
5. Introduction
Primary Sensing
Usually treated using classical detection theory.
The decision is made among two hypothesis:
H0 : xn = wn , n = 1, 2, . . . , N
(1)
H1 : xn = wn + sn , n = 1, 2, . . . , N
Neyman-Pearson (N-P) test statistic:
fH1 (x )
L (x ) = , (2)
fH0 (x )
where fH (x ) is the joint pdf of the observed samples for hypothesis H
Provides optimal detection if pdfs in (2) are known.
Some famous detectors: Energy detector, Cyclostationary detectors, CAV
detectors, Corrsum, and others.
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
6. Noise Uncertainty
Given a perfect noise information, detection is possible at any SNR with
energy detector.
Practical systems will only have a estimate of the noise variance σ2 . This
imperfect knowledge is refereed to as noise uncertainty (NU).
The NU concept was identified and studied in detail in [2].
In [2], σ2 is assumed to be confined in the interval [σ2 , σ2 ], but otherwise
lo hi
unknown.
Worst-case detector assumes
σ2 under H0
σ2 = hi
σ2
lo under H1
For some value of SNR, the detector exhibits Pd < Pfa , regardless of the
number of samples ⇒ SNR wall
Below SNR wall, no useful detection is possible for the model above.
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
7. Noise Uncertainty
So, the main idea behind this work is to find out a good statistical model for the
NU and investigate if we can avoid the SNR wall be detailed statistical
modeling.
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
8. System Model
General Assumptions
Define random noise parameter α = 1/σ, where σ2 is the variance.
Assume noise/signal Gaussian
α
f (xn |α) = √ exp{−α2 xn /2},
2
(3)
2π
Assuming i.i.d. process, the marginal pdf of sample vector x is
N
1 ∞ α2
f (x ) = f (α)αN exp − ∑ xn2 d α, (4)
(2π)N /2 0 2 n =1
where f (α) is the distribution of the noise parameter α
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
9. System Model
Noise Uncertainty Model
Popular Log Normal Model:
1 1
fLN (α) = √ exp − (log α + µLN )2 /σ2
LN (5)
ασLN 2π 2
Fit to truncated Gaussian with
µ = E {α} = exp{−µLN + σ2 /2},
LN (6)
2 2
σ = Std{α} = [exp(σLN ) − 1] exp(−2µLN + σLN ), (7)
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
10. System Model
Noise Uncertainty Model
Log Normal vs. Gaussian Approximation
LogNorm
6 NU = 0.5 dB Gauss
f (α)
4
2
0
0.7 0.8 0.9 1 1.1 1.2 1.3
α
NU = 1.0 dB LogNorm
3
Gauss
f (α)
2
1
0
0.6 0.8 1 1.2 1.4 1.6
α
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
11. Detection with NU
Case I: Uncorrelated Signal Samples
2
In (4), see that p = ∑n xn sufficient statistic.
Pdf of p conditioned on noise parameter
α2
f (p|α) = (α2 p)N /2−1 exp{−α2 p/2}, (8)
2N /2 Γ(N /2)
Required marginal distribution on p only:
1 ∞ α2 p
f (p)= f (α)α2 (α2 p)N /2−1 exp{− } d α. (9)
2N /2 Γ(N /2) 0 2
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
12. Detection with NU
Case I: Uncorrelated Signal Samples
Using the Gaussian model for f (α), we can derive the closed-form f (p)
as follows:
c0 e−c3 N
N k
c2
f (p)= ∑ L
Γ(Lk ) 1+(−1)N −k Γ Lk , c1 c2
2
(10)
2 k =0 k c1 k
where Lk = (N + 1 − k )/2 and
pN /2−1 p 1
c0 = √ c1 = +
2N /2 Γ(N /2) 2πσα 2 2σ2
1 µ2 1
c2 = µα /(σ2 p + 1)
α c3 = α
2
1− 2
2 σα σα p + 1
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
13. Detection with NU
Case I: Uncorrelated Signal Samples
Example Detection Performance
Parameters: SNR=0 dB, NU=1 dB, N = 20 samples
Proposed detector knows σα but not realizations of α
For robust (worst-case) detector let α ∈ [µα − 1.5σα , µα + 1.5σα ]
1
0.9
0.8
0.7
0.6
Pd
0.5
0.4
0.3
0.2 Modeled NU
0.1 Worst Case NU
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Pfa
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
14. Detection with NU
Case II: Correlated Signal Samples
Assume a correlated primary user signal with a covariance matrix Σs .
Consider the following assumptions:
s
´
Σs = σ2 .Σs , where σ2 is the signal variance.
s
σ2 = σ2 .γ, where σ2 is the noise variance and γ is the SNR.
s
SNR is constant, one can think about it to be the worst SNR.
Then, the marginal pdfs of the received signal for both hypothesis are:
H0
1 ∞ α2
f (x ) = f (α)αN exp − XT X d α, (11)
´
(2π)N /2 |Σs + I |
1/
2 0 2
H1
1 ∞ α2
f (x ) = f (α)αN exp − XT (Σs + I )−1 X d α,
´
´
(2π)N /2 |Σs + I|
1/
2 0 2
(12)
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
15. Detection with NU
Case II: Correlated Signal Samples
Now, consider the following:
Make integration for the exponential parts since they assume to have the
most effect.
Take the Eigendecomposition of the signal covariance matrix.
Then, the N-P detector can be derived as follow:
µ2
2 2 erfc − α
2σ2 (1+σ2 B1 )
µα B0 µα B1 1 + 2σ2 B α 0 α α
L(Y) = exp −
1 + 2σ2 B0
α 1 + 2σ2 B1
α 1 + 2σ2 B1
α µ2
erfc − α
2σ2 (1+σ2 B0 )
α α
(13)
where
Y is the uncorrelated version of the received signal X with
σ2 I for H0
Σy =
σ2 (γΛ + I ) for H1
´
where Λ = diag (λ1 ...λN ) and λn is the nth eigenvalue of Σs
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
16. Detection with NU
Case II: Correlated Signal Samples
Continue ..
B0 = 1 ∑N=1 yn and B1 = 1 ∑N=1 λan yn
2 n
2
2 n
2
´
where λan is the nth eigenvalue of the matrix A = (γΣs + I )−1
Using the identity (Q + ρM)−1 Q − ρQ−1 MQ−1 , we have
´
A I − γ.Σs . Note this identity is used for small values of γ
Then, B1 B0 − 2 γ. ∑N=1 λn yn = B0 − R
1
n
2
Note B0 represents the signal energy, where R is seen to be a
correlation-based value.
Taking the logarithm of the NP detector in (13), rewriting it in terms of B0
and R and considering only the exponential term:
µ 2 B0
α µ2 B0 − µ2 R
α α
l (y ) = − (14)
1 + 2σ2 B0 1 + 2σ2 B0 − 2σ2 R
α α α
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
17. Detection with NU
Case II: Correlated Signal Samples
Assuming a covariance matrix with an exponential correlation model, as
follows:
1 for i = j
cov (xi , xj ) = σ2 .γ.
ρ|i −j | for i = j
i , j = 1, 2, .., N and ρ is the correlation coefficient
The inverse of the covariance matrix is then known to be tridiagonal matrix,
and the a closed form for the eigenvalues of this tridiagoal matrix can be
obtained. Then, a closed form for the eigenvalue λn could be as follows:
γ.(1 − ρ2 )
λn = (15)
1 + ρ2 + 2ρ cos( Nπn1 )
+
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
18. Detection with NU
Case II: Correlated Signal Samples
At this point, I don’t have clear results to show for the next steps. I trying to
study more the detector in (14) by applying Taylor series expansion and
performing sensitivity analysis to investigate how dominant B0 and R are for
with respect to the number of samples, NU level and SNR
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
19. Noise Calibration Measurment
Since most of the noise in a true receiver comes from the front-end LNA, a
simple architecture depicted below can be used for noise calibration
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
21. Conclusion
Noise uncertainty limits robust detection at low SNR.
SNR can be relaxed by simple NU modeling.
Experiment demonstrates useful detection to -16 dB
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
22. Future Work
More analysis on the detector in case of a correlated signal
Study the importance of signal energy and correlation-based value on the
detection in case of a correlated signal.
Make more measurements with longer integration times and lower grade
amplifiers.
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
23. Published Work
Hammouda, M. and Wallace, J., ”Noise uncertainty in cognitive radio sensing:
analytical modeling and detection performance,”, the 16th International ITG
Workshop on Smart Antennas WSA,2012
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio
24. References
Mitola, J., III and Maguire, G. Q., Jr., ”Cognitive radio: Making software radios
more personal,”, IEEE Personal Commun. Magazine, vol. 6, pp. 1318, Aug. 1999.
R. Tandra and A. Sahai, ”SNR walls for signal detection,”, IEEE J. Selected
Topics Signal Processing, vol. 2, pp. 417, Feb. 2008. 1318, Aug. 1999.
S. M. Kay, ”Fundamentals of Statistical Signal Processing: Detection Theory,”,
Prentice Hall PTR, 1998.
F. Heliot, X. Chu, and R. Hoshyar, ”A Tight closed-form approximation of the
Log-Normal fading channel capacity,”, IEEE Transaction on Eireless
Communications, vol. 8, No. 6 , June. 2009. 1318, Aug. 1999.
Marwan A. Hammouda Noise Uncertainty in Cognitive Radio