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In this lab you are going to apply the Fast Fourier Transform ( FFT ) to analyze the spectral content of an input signal in real time. You will alsoexplore algorithms that estimate a stationary random signal's Power Spectral Density ( PSD ). Finally, you will be introduced to using the C environment and codeoptimization in a practical application. This knowledge will be applied in optimizing a reference implementation of a PSD estimator.
First, samples of the power spectrum of a deterministic signal will be calculated via the magnitude squared of the FFT of the windowedsignal. You will transform a 1024-sample block of input data and send the power spectrum to the output for display on theoscilloscope. After computing the FFT of a 1024-sample block of input data, you will thencompute the squared magnitude of the sampled spectrum and send it to the output for display on the oscilloscope. In contrastto the systems you have implemented in the previous labs, the FFT is an algorithm that operates on blocks of samples at atime. In order to operate on blocks of samples, you will need to use interrupts to halt processing so that samples can betransferred.
The FFT can be used to analyze the spectral content of a signal. Recall that the FFT is an efficient algorithm forcomputing the Discrete Fourier Transform ( DFT ), a frequency-sampled version of the DTFT .
DFT:
Your implementation will include windowing of the input data prior to the FFT computation. This is simple a point-by-pointmultiplication of the input with an analysis window. As you will explore in the prelab exercises, the choice of windowaffects the shape of the resulting spectrum.
A block diagram of the spectrum analyzer you will implement in the lab, including the required input andouput locations, is depicted in .
Second, you will generate a colored, psuedo-noise (PN) sequence as input to the power spectrum algorithm. The noise sequence will begenerated with a linear feedback shift register, whose operation is as shown in . This PN generator is simply a shift-register and an XOR gate. Bits 0, 2, and 15 of the shift-registerare XORed together and the result is shifted into the lowest bit of the register. This lowest bit is the output of the PN generator,and the highest bit is discarded in the shift process. The LSB is used to generate avalue of $M$ and this sequence will repeat with a period of $2^{B}-1$ , where $B$ is the width in bits of the shift register and $M$ is a constant. The power spectral density of this sequence is flat(white) and it will be "colored" via a fourth-order IIR filter. PN generators ofthis type are a useful source of random data bits for system testing. They are especially useful as a data model in simulating communicationsystems as these systems tend to randomize the bits seen by the transmission scheme so that bandwidth can be efficiently utilized.However, this method will not produce very "random" numbers. For more on this, see Pseudorandom number generator , Linear feedback shift register , and chapter 7, section 4 of Numerical Recipes .
The direct-power-spectrum (DPS) algorithm outlined above is insufficient for estimating the PSD of a stationary noisesignal because the variance of the estimated PSD is proportional to the value of the actual PSD. For the third part of this lab you will try toreduce the variance of the PSD estimate by windowing the autocorrelation of the noise signal and computing the fft.
The autocorrelation of a sequence is the correlation of the sequence with itself:
For random signals, the autocorrelation here is an estimate of the actual autocorrelation.As $\left|m\right|$ is increased, the number of samples used in the autocorrelation decreases.The windowed-DPS algorithm is equivalent to taking the FFT of the autocorrelation of the windowed data. Windowing of the dataadds even more noise to the autocorrelation estimate, since the autocorrelation is performed on a distorted version of the originalsignal. An improvement can be made by constructing an accurate estimate of the autocorrelation (using a rectangular window),applying a window and computing the FFT. The motivation for applying the window at the latter stage is that emphasis should be given toaccurate autocorrelation values while less accurate values should be de-emphasized or discarded.
A good empirical characterization of a random process requires sufficient data, and both of the PSD-estimation algorithms definedabove can be extended to accomodate more data. There is one caveat, however: manyreal-world processes are modeled as short-time stationary processes (non-stationary models are hard to deal with), so there is a practical limit to how much data is available for a PSDestimate. Additional data is added to the direct-PSD estimation algorithm byadding multiple spectra together, thereby smoothing the PSD estimate. Additional data is added to the windowed-autocorrelation methodby computing the autocorrelation of the total data set before windowing. You will explore the windowed-autocorrelation method onthe DSP.
A second objective of this lab exercise is to introduce the TMS320C549 C environment in a practical DSP application. The Cenvironment provides a fast and convenient way to implement a DSP system using C and assembly modules. You will also learn howto optimize a program and when to use C or assembly. In future labs, the benefits of using the C environment will become clear as largersystems are developed.
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