Comment on the effect of the measurement noise on the

Question

Design of Bandpass Sampling for Spectral Analysis

The project report must be typed and a printed copy submitted to the instructor. The report should include sufficient detail to describe your design methodology and the resulting system performance. The basic "rule of thumb" for the report is that it should contain enough detail that if you were to read it in a year or two, you could easily follow the design methodology, understand the results presented, and be able to reproduce the design and results with relatively little effort.

Design a bandpass sampling system and use it to analyze a bandpass signal. As motivation, consider the problem of analyzing the spectrum of a narrowband signal (of bandwidth at most 4 MHz) within the [1.993, 1.997] GHz range. For the design exercise, we scale the problem down by a factor of 1000, so the task is to analyze a 4 kHz bandwidth signal within the [1.993, 1.997] MHz frequency band. Design the bandpass sampling scheme so that spectral analysis of the signal can have frequency resolution of 20 Hz (or less).

Begin with the bandpass signal x(t) = 5 sin(2Πf_at) + 1000 cos(2Πf_bt) - cos(2Πf_ct) where, for the problem, f_a = 1904015 H_z; f_b = 1905193 H_z; f_c = 1905635 H_z. Note that the signal consists of sinusoids of very different amplitudes, all fairly close together. Part of the task is to use spectral analysis to resolve the different frequencies. For the (Matlab) numerical work, sample this signal at rate 8 MHz and acquire 500,000 samples, but with additive white Gaussian noise samples of variance σ² added to each sample (to model the realistic case of measurement noise).

The sampled signal is then

x(n) = 5 sin(2Πf_anT) + 1000 cos(2Πf_bnT) - cos(2Πf_cnT) + w(n) ; n = 0, ... ,499999

For the exercise, try three different noise powers, σ_w = 0, 1, and 10. This signal forms the bandpass signal to use in the simulations. Using this discrete-time signal, compute the DFT (use the FFT function in Matlab) and plot the magnitude (in decibels). Can you distinguish between the different sinusoids in the signal? What effect does changing the noise power have on your ability to observe the different sinusoids?

Design a suitable bandpass sampling scheme; specifically, determine lower and upper frequencies for the bandpass frequency band and determine the bandpass sampling rate. A convenient design will use a bandpass sampling rate so that the ratio of the original 8 MHz sampling rate to the bandpass sampling rate is an integer.

Design a linear phase, FIR filter to bandpass filter the signal prior to bandpass sampling. Do simulations for the case of bandpass filtering first and then bandpass sampling, as well as no filtering and just bandpass sampling. From the signal sampled at 8 MHz, form the bandpass sampled signal, say x_bp(n). Compute the DFT (so that the frequency resolution is 20 Hz or less) and plot the magnitude spectrum (in decibels). Properly adjust the horizontal axis in your plot to correspond to the bandpass frequency range (in Hz). Can you resolve the different sinusoids?

Finally, use at least two different windows prior to computing the DFT (e.g., the Hanning and Blackman windows- note that the window signal should be DFT-even, so use the ‘periodic' option with the Matlab window function, e.g., w_hanning = hanning(N, ‘periodic');).

Comment on the effect of the measurement noise on the results, especially for the cases of no bandpass filtering before bandpass sampling, and with bandpass filtering before bandpass sampling. Explain.