1==============What is aliasing and what techniques are used to avoid it? Use ex

Question

1==============What is aliasing and what techniques are used to avoid it? Use examples to illustrate your answers and be sure to cite references.

2=============Use the Internet to research the kind of difference equation that can be used to generate an echo from a recorded sequence of sounds. Be sure to cite your references.

3===============What are the advantages and disadvantages of using NICAD, NIMH, lithium, or sealed lead-acid rechargeable batteries?

4===============Compare the power requirements for the subsystems of your project to the power requirements for the motors.

Explanation / Answer

In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or aliases of one another) when sampled. It also refers to the distortion or artifact that results when the signal reconstructed from samples is different from the original continuous signal.

Aliasing can occur in signals sampled in time, for instance digital audio, and is referred to as temporal aliasing. Aliasing can also occur in spatially sampled signals, for instance digital images. Aliasing in spatially sampled signals is called spatial aliasing. Assume that a digital data set, representing a signal sample value sequence, is given. It can be shown graphically,These signal samples might be used for reconstruction of the original analog signal they belong to. Aliasing results in an uncertainty. Indeed, look at these frequencies. Which of the shown frequencies is the right one? Evidently impossible to say if some additional information is not supplied.

The effect of aliasing, of course, is well known. As well as the means how to avoid it. It has been proved that there will be no uncertainty if all frequencies above a certain level are filtered off the original signal by a lowpass filter. That is actually what is required by the famous Sampling Theorem. To sample at a frequency at least twice higher than the highest frequency present in the signal spectrum. However, while the satisfaction of this requirement resolves the aliasing problem, such compulsory filtering off of the signal upper frequencies also imposes a very drastic limitation on application of digital signal processing in frequency domain. The achievable bandwidth of digital systems then is determined by the achievable sampling rate. The latter obviously depends on the microelectronic techniques used. To widen the bandwidth in order to cover higher frequencies, more costly and more power consuming chips have to be used.

It is very tempting to do something and to eliminate this limitation. That would open up a broad area of new exciting digital signal processing applications. Apparently, if there would be some other way how to avoid aliasing, digital processing of signals would be applicable in a much broader frequency range. So the question is: is there such a possibility? Fortunately, the answer is affirmative. Application of nonuniform sampling offers this.

To see how nonuniform sampling helps in avoiding aliasing, The lower frequency sine function is again sampled and the corresponding data set is obtained. However the intervals between the sampling instants now are not equally long. The distances between the sampling instants along the time axis differ. And that proves to be very useful. Indeed, it can be easily seen that only one sine function can be drawn exactly through the indicated points representing the sample value sequence taken from the first sinusoid. Other frequencies simply do not go through them.

This effect can be easily checked. Studies of it would confirm the fact that, in the case of correctly performed nonuniform sampling, each frequency is represented by a unique data set. Therefore nonuniformly sampled signals have no completely overlapping aliases like those observed at periodic sampling. Consequently, it can be expected that application of nonuniform sampling should open up the possibility of distinguishing all frequencies present in the signal spectrum, even those substantially exceeding the mean sampling rate.

The equation that describes the simple echo case is,
r(t) = s(t- t1) + a s([t- t1] + td )
In effects processing the common transit delay t1 is often
ignored since it is the relative delay td that will determine the
echo effect.

A more practical alternative is to limit the bandwidth of the signal below one-half the sample rate with a low-pass or anti-alias filter, which can be implemented on each input channel in front of the A/D converter. Low-pass filtering must be done before the signal is sampled or multiplexed, since there is no way to retrieve the original signal once it has been digitized and aliased signals have been created.

To avoid aliasing with a low-pass filter, two processes actually must occur:

As dictated by the Nyquist theory (see opposite page), the input signal must be sampled at a rate of at least twice the highest frequency component of interest within the input signal.

Any frequency components above half the sampling rate (also called the Nyquist frequency) must be eliminated by an anti-alias filter before sampling. Under ideal conditions, a low-pass filter would exactly pass unchanged all slower signal components with frequencies from DC to the filter cutoff frequency. Faster components above that point would be totally eliminated, reducing the signal disturbance. But, real filters do not cut off sharply at an exact point. Instead, they gradually eliminate frequency components and exhibit a falloff or rolloff slope. These attenuation slopes typically range from 45 dB/octave to 120 dB/octave and "bottom out" at some finite value of stopband rejection, typically 75 to 100 dB.

A simple illustration of these processes can be seen in the case of the 800 Hz frequency aliasing to 200 Hz (see above). Suppose that the 800 Hz is an unwanted interfering signal caused by an unwanted mechanical vibration. To prevent its alias from causing significant data errors at 200 Hz, the 800 Hz frequency must be removed by a low-pass filter. If the cutoff point is set near 450 Hz, a filter with a steep rolloff slope will eliminate the 800 Hz frequency, making the false 200 Hz frequency disappear. The input frequencies of interest below the filter cutoff (450 Hz) will still pass through the system unchanged.

High-frequency components can result from the inherent noise of the system itself and from noise or interference not related to the DAS, including 50Hz or 60Hz pickup, broadcasting stations, and mechanical vibrations. High-frequency components also are inherent in any sharp transitions of the measured signal. Low-pass filters generally can eliminate alias errors produced by these sources as long as the filters precede the A/D converter.

Benefits of Using Low-Pass Filters

The aliasing phenomenon becomes a problem in A/D conversion systems when an input signal contains frequency components above half the A/D sampling rate. These higher frequencies can "fold over" into the lower frequency spectrum and appear as erroneous signals that cannot be distinguished from valid sampled data. The best approach to eliminating false lower frequencies is to use a low-pass filter, which inhibits aliasing by limiting the input signal bandwidth to below half the sampling rate. A low-pass filter, which is applied to each input channel in front of the A/D card, also eliminates unwanted high-frequency noise and interference introduced prior to sampling. It reduces system cost, acquisition storage requirements, and analysis time by allowing for a lower sampling rate. Finally, a low-pass filter serves as an important element of any data acquisition system in which the accuracy of the acquired data is essential.

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