1 ? μ μ 1 σ 1 parenrightbigg where q 1 x is the

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1 ( ǫ ) + μ 0 - μ 1 σ 1 parenrightbigg where Q 1 ( x ) is the inverse Q function. The Q ( · ) and Q 1 ( · ) functions are available in MATLAB as qfunc.m and qfuncinv.m . 3. Suppose that the signal vector is zero-mean Gaussian, but has time-varying variance given by σ 2 s ( n ). Simplify the LLR defined in Section 1.3.1 to show that the optimal detector is a weighted energy detector. 4. If x(t) is wide-sense cyclostationary, its mean and autocorrelation are periodic. Express R x ( t - τ 2 , t + τ 2 ) as a Fourier series. If x(t) is cyclostationary, what does that imply about frequency components at m/T 0 , where m is an integer? 5. Given Eq. (8) and using the Einstein-Wiener-Khintchine (EWK) relation, derive the spectral autocorrelation function of a wide sense cyclostationary process. 8
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2 Software Implementation 2.1 Creating the Transmission Environment Open the datagen.mdl file available on the course website. This model features three very basic modulation schemes that are pulse shaped for over the air transmission. Vectors of each transmission are saved to the MATLAB workspace. Begin by observing the output of each channel on an fftscope block. Next vary the SNR of the channel. At what point does the signal become unobservable due to noise? Figure 6: Data generation model. Run the model shown in Figure 6 once and go to the MATLAB workspace. 2.2 Energy Detector Construction We will now proceed with the construction of a simple energy detector that analyzes the signals in the workspaces and determines whether or not a signal is present based on a threshold. Recall that an energy detector uses the energy spectra of the received signal in order to identify the frequency locations of the transmitted signal. As a result, the following steps that are also illustrated in Figure 7 will assist in producing the frequency representation of the intercepted signal: 1. Pre-filtering of intercepted signal extracts frequency band of interest. 2. Analog-to-digital conversion (ADC) converts filtered intercepted signal into discrete time sam- ples. 9
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3. Fast Fourier transform (FFT) provides the frequency representation of the signal. 4. Square-law device yields the square of the magnitude of the frequency response from the FFT output. A/D (.) 2 Averages N samples Pre-Filter FFT Figure 7: Schematic of an energy detector implementation employing pre-filtering and a square-law device. To perform the actual energy detection, we need to apply a threshold in the frequency domain, which is used to decide whether a transmission is present a specific frequency. As a result, any portion of the frequency band where the energy exceeds the threshold is considered to be occupied by a transmission since energy detection is a binary decision making process consisting of two hypotheses: H 0 (idle) or H 1 (occupied). However, one of the primary challenges of energy detection is the selection of an appropriate threshold. This is due to the fact that the threshold may work for one transmission
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