Frequencies: Which one is the aliasing frequency? Estimate the aliasing noise le

Question

Frequencies:


Which one is the aliasing frequency? Estimate the aliasing noise level:



% aliasing noise level=(Peak RMS volatage)/(Input RMS voltage)=(Peak RMS volatage)/(0.707 volts)=

Compare the result to the one calculated using the following formula:

% aliasing noise level=?(1+(f_a/f_C )^2n )/?(1+((f_S-f_a)/f_C )^2n )=


where f_S= sampling frequency, f_C= cut-off frequency, and f_a= aliasing frequency.

From the second spectrum analyzer:

Frequencies:

Explain:

Explanation / Answer

When a digital image is viewed, a reconstruction—also known as an interpolation—is performed by a display or printer device, and by the eyes and the brain. If the resolution is too low, the reconstructed image will differ from the original image, and an alias is seen. An example of spatial aliasing is the Moiré pattern one can observe in a poorly pixelized image of a brick wall. Techniques that avoid such poor pixelizations are called anti-aliasing. Aliasing can be caused either by the sampling stage or the reconstruction stage; these may be distinguished by calling sampling aliasing prealiasing and reconstruction aliasing postaliasing.[1] Temporal aliasing is a major concern in the sampling of video and audio signals. Music, for instance, may contain high-frequency components that are inaudible to humans. If a piece of music is sampled at 32000 samples per second (sps), any frequency components above 16000 Hz (the Nyquist frequency) will cause aliasing when the music is reproduced by a digital to analog converter (DAC). To prevent that, it is customary to remove components above the Nyquist frequency (with an anti-aliasing filter) prior to sampling. But any realistic filter or DAC will also affect (attenuate) the components just below the Nyquist frequency. Therefore, it is also customary to choose a higher Nyquist frequency by sampling faster. In video or cinematography, temporal aliasing results from the limited frame rate, and causes the wagon-wheel effect, whereby a spoked wheel appears to rotate too slowly or even backwards. Aliasing has changed its apparent frequency of rotation. A reversal of direction can be described as a negative frequency. Temporal aliasing frequencies in video and cinematography are determined by the frame rate of the camera, but the relative intensity of the aliased frequencies is determined by the shutter timing (exposure time) or the use of a temporal aliasing reduction filter during filming.[2] Like the video camera, most sampling schemes are periodic; that is they have a characteristic sampling frequency in time or in space. Digital cameras provide a certain number of samples (pixels) per degree or per radian, or samples per mm in the focal plane of the camera. Audio signals are sampled (digitized) with an analog-to-digital converter, which produces a constant number of samples per second. Some of the most dramatic and subtle examples of aliasing occur when the signal being sampled also has periodic content. Sinusoids are an important type of periodic function, because realistic signals are often modeled as the summation of many sinusoids of different frequencies and different amplitudes (with a Fourier series or transform). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum. Two different sinusoids that fit the same set of samples. Here a plot depicts a set of samples whose sample-interval is 1, and two (of many) different sinusoids that could have produced the samples. The sample-rate in this case is f_s, = 1. For instance, if the interval is 1 second, the rate is 1 sample per second. Nine cycles of the red sinusoid and 1 cycle of the blue sinusoid span an interval of 10. The respective sinusoid frequencies are f_mathrm{red}, = 0.9 and f_mathrm{blue}, = 0.1. In general, when a sinusoid of frequency f, is sampled with frequency f_s,, the resulting samples are indistinguishable from those of another sinusoid of frequency scriptstyle (f - Nf_s),, for any integer N. The values corresponding to N ? 0 are called images or aliases of frequency f., In our example, the N=±1 aliases of scriptstyle f = f_mathrm{red} = 0.9 are scriptstyle 0.9 + 1.0 = 1.9 and scriptstyle 0.9 - 1.0 = -0.1. A negative frequency is equivalent to its absolute value, because sin(-wt+?)=sin(wt-?+p), and cos(-wt+?)=cos(wt-?). Therefore we can express all the image frequencies as f_mathrm{image}(N) = |f - Nf_s|,, for any integer N (with scriptstyle f_mathrm{image}(0) = f, being the actual signal frequency). Then the N=1 alias of scriptstyle f_mathrm{red}, is scriptstyle f_mathrm{blue},, (and vice versa). Aliasing matters when one attempts to reconstruct the original waveform from its samples. The most common reconstruction technique produces the smallest of the scriptstyle f_mathrm{image}(N), frequencies. So it is usually important that scriptstyle f_mathrm{image}(0), be the unique minimum. A necessary and sufficient condition for that is scriptstyle f_s/2 > |f|,, where scriptstyle f_s/2, is commonly called the Nyquist frequency of a system that samples at rate scriptstyle f_s., In our example, the Nyquist condition is satisfied if the original signal is the blue sinusoid (scriptstyle f = f_mathrm{blue}). But if scriptstyle f = f_mathrm{red} = 0.9, the usual reconstruction method will produce the blue sinusoid instead of the red one. Folding The black dots are aliases of each other. The solid red line is an example of adjusting amplitude vs frequency. The dashed red lines are the corresponding paths of the aliases. As scriptstyle f increases from 0 to scriptstyle f_s/2, scriptstyle f_mathrm{image}(1) goes from scriptstyle f_s to scriptstyle f_s/2. Similarly, as scriptstyle f increases from scriptstyle f_s/2 to scriptstyle f_s, scriptstyle f_mathrm{image}(1) continues decreasing from scriptstyle f_s/2 to 0. A graph of amplitude vs frequency for a single sinusoid at frequency scriptstyle 0.6 f_s and some of its aliases at scriptstyle 0.4 f_s, scriptstyle 1.4 f_s, and scriptstyle 1.6 f_s would look like the 4 black dots in the adjacent figure. The red lines depict the paths (loci) of the 4 dots if we were to adjust the frequency and amplitude of the sinusoid along the solid red segment (between scriptstyle f_s/2 and scriptstyle f_s). No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and scriptstyle f_s. This symmetry is commonly referred to as folding, and another name for scriptstyle f_s/2 (the Nyquist frequency) is folding frequency. Folding is most often observed in practice when viewing the frequency spectrum of real-valued samples using a discrete Fourier transform. Complex sinusoids Complex sinusoids are waveforms whose samples are complex numbers, and the concept of negative frequency is necessary to distinguish them. In that case, the frequencies of the aliases are given by just: f_mathrm{image}(N) = f - Nf_s., Therefore, as scriptstyle f, increases from scriptstyle f_s/2, to scriptstyle f_s,, scriptstyle f_mathrm{image}(1) goes from scriptstyle -f_s/2, up to 0. Consequently, complex sinusoids do not exhibit folding. Complex samples of real-valued sinusoids have zero-valued imaginary parts and do exhibit folding.

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