|
|
|
|
|
ALIASING AND SAMPLING USING ORTHONORMAL BASES
|
|
|
Instructional report by Yogesh Sawant (ID: 104605545) in partial fulfillment of course requirements of ESE 558 Submitted to Professor Murali Subbarao Aliasing Aliasing is a potential problem whenever an analog signal is point sampled to convert it into a digital signal Aliasing is what happens when analog data is represented on a digital system. It happens whenever an analog signal is not sampled at a high enough frequency. A curved line drawn on a grid, where the curved line represents the analog data and the grid represents the digital system, is a good example of analog data on a digital system (see figure 1.1) Figure 1.1 Figure 1.2 Digital version of the line Line on grid representing analog data on digital system When the analog data is converted to digital some problems arise. The digital system in this example is the grid. To convert the analog line to a digital line each point in the grid may either represent a point in the line, by being filled in, or represent an area where the line does not exist, by remaining white. There can't be a square that is only partly filled. Each square must be either filled in or not. In other words, to draw the line in digital format we need to completely fill in any square that a portion of the line passes through. That's all part of it being digital. Okay no problem, right? The line goes through the different squares so we'll fill in each square that the line goes through. Figure 1.2 shows what the line looks like when we do this. Not very smooth, is it? We no longer have curves; all we have is a choppy line made up of squares and rectangles. Now let us describe the same thing discussed above in slightly technical terms. A signal x(t), periodically sampling at time instances t=kTs, produces a time-series x[k]={x[0], x[Ts], x[2Ts], …}. Consider the example shown below which consists of sinusoid and an added "glitch." Prior analysis, using a spectral analyzer failed to detect the low-energy broadband (viz., spectrum) glitch since it is overwhelmed by the energy in the sinusoid. Upon sampling, the glitch is missed altogether (as shown) and after reconstruction a pure sinusoid is recovered. The proud engineer is left with the belief that Shannon's Sampling Theorem has been successfully applied to the problem. The system is delivered to customer who immediately complains that the "anomaly" detector he purchased doesn't work reliably. The customer says that he can only detect a few random occurrences of the anomaly (glitch) every 100ms when there should have been over 100. The engineer, in order to solve the above problem, designs another system with a sampling rate 100 times faster than the first, does this fix the problem? This is an error due to aliasing. The frequency response of the glitch may be 10x, 100x or greater than the Nyquist rate defined by the sinusoid. What needs to be done is to establish a mechanism by which these errors can be predicted and quantified. Consider the data shown below which illustrates a process resulting in an aliasing error. In the diagram, the high frequency signal is sampled just under the Nyquist rate. As a result, each sample is taken at a slightly later in each cycle.
|
|
|
|
Still Can't Find What Your Looking For? Then Try a Essay Search!
|
