# Spectral densities of filtered processes

It only takes a minute to sign up. I need a change of variables here. The final answer seems correct but I can't help but feel that I skipped an important argument in the derivation. Could someone please take a look at this and make it rigorous if needed? Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.

It only takes a minute to sign up. I'm confused by this formula since up to now I thought the power spectral density is given by the signal mutliplied with the absolute value squared of the transfer function. But why do I have to consider the expectation value in this case and why am I scaling by the sampling time?

Sign up to join this community. The best answers are voted up and rise to the top. What is the power spectral density after filtering? Ask Question. Asked 2 years, 11 months ago. Active 2 years, 11 months ago. Viewed times. Thx for any help! MBaz I'm not sure what you mean, but you're welcome to change whatever you feel necessary. See also dsp. Not sure if this is random. Bit errors occur random. Of course, you can also find the PSD for a specific, deterministic sequence, but then the result would only be valid for that one case.

Expert's Answer Solution. Feedback :. Next Previous. Related Questions. Derive and sketch the actual spectral density function for Xt Derive and sketch the actual spectral density function for Xt.

Find the modulated signal y tthe filtered signal yf tthe numerical value of the For this problem If that signal is sampled at samples per second, write the expression x n of the Consider data on the weights of cork borings in four directions north, east, south, west for Consider data on the weights of cork borings in four directions north, east, south, west for 28 trees in a block of plantations see Exercise 3.

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Click here to Login. No Account Yet? Click here to Sign Up. Sign in with Facebook.The statistical average of a certain signal or sort of signal including noise as analyzed in terms of its frequency content, is called its spectrum. When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density.

More commonly used is the power spectral density or simply power spectrumwhich applies to signals existing over all time, or over a time period large enough especially in relation to the duration of a measurement that it could as well have been over an infinite time interval. The power spectral density PSD then refers to the spectral energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite.

For instance, the pitch and timbre of a musical instrument are immediately determined from a spectral analysis. Obtaining a spectrum from time series such as these involves the Fourier transformand generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a dispersive prism is used to obtain a spectrum of light in a spectrographor when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency.

However this article concentrates on situations in which the time series is known at least in a statistical sense or directly measured such as by a microphone sampled by a computer. The power spectrum is important in statistical signal processing and in the statistical study of stochastic processesas well as in many other branches of physics and engineering.

Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of spatial frequency. Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. When these signals are viewed in the form of a frequency spectrum, certain aspects of the received signals or the underlying processes producing them are revealed.

In some cases the frequency spectrum may include a distinct peak corresponding to a sine wave component.

### Spectral density

And additionally there may be peaks corresponding to harmonics of a fundamental peak, indicating a periodic signal which is not simply sinusoidal.

Or a continuous spectrum may show narrow frequency intervals which are strongly enhanced corresponding to resonances, or frequency intervals containing almost zero power as would be produced by a notch filter.

In physicsthe signal might be a wave, such as an electromagnetic wavean acoustic waveor the vibration of a mechanism. The power spectral density PSD of the signal describes the power present in the signal as a function of frequency, per unit frequency. When a signal is defined in terms only of a voltagefor instance, there is no unique power associated with the stated amplitude. In this case "power" is simply reckoned in terms of the square of the signal, as this would always be proportional to the actual power delivered by that signal into a given impedance.

But it is mathematically preferred to use the PSD, since only in that case is the area under the curve meaningful in terms of actual power over all frequency or over a specified bandwidth. Here g denotes the g-force.

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Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of x t will remain unspecified, but the independent variable will be assumed to be that of time. Energy spectral density describes how the energy of a signal or a time series is distributed with frequency.

The energy spectral density is most suitable for transientsâ€”that is, pulse-like signalsâ€”having a finite total energy. In this case, Parseval's theorem [9] or Plancherel's theorem gives us an alternate expression for the energy of the signal:. The above definition of energy spectral density is suitable for transients pulse-like signals whose energy is concentrated around one time window; then the Fourier transforms of the signals generally exist.

For continuous signals over all time, such as stationary processesone must rather define the power spectral density PSD ; this describes how power of a signal or time series is distributed over frequency, as in the simple example given previously. Here, power can be the actual physical power, or more often, for convenience with abstract signals, is simply identified with the squared value of the signal. Note that a stationary processfor instance, may have a finite power but an infinite energy.

After all, energy is the integral of power, and the stationary signal continues over an infinite time. That is the reason that we cannot use the energy spectral density as defined above in such cases.The Whittakerâ€”Shannon interpolation formula or sinc interpolation is a method to construct a continuous-time bandlimited function from a sequence of real numbers. The formula dates back to the works of E.

Borel inand E. Whittaker inand was cited from works of J.

## Spectral density

Whittaker inand in the formulation of the Nyquistâ€”Shannon sampling theorem by Claude Shannon in It is also commonly called Shannon's interpolation formula and Whittaker's interpolation formula. Whittaker, who published it incalled it the Cardinal series. When the sampled function has a bandlimit, Bless than the Nyquist frequency, x t is a perfect reconstruction of the original function.

See Sampling theorem. Otherwise, the frequency components above the Nyquist frequency "fold" into the sub-Nyquist region of X fresulting in distortion. See Aliasing. The interpolation formula is derived in the Nyquistâ€”Shannon sampling theorem article, which points out that it can also be expressed as the convolution of an infinite impulse train with a sinc function :.

This is equivalent to filtering the impulse train with an ideal brick-wall low-pass filter with gain of 1 or 0 dB in the passband. If the sample rate is sufficiently high, this means that the baseband image the original signal before sampling is passed unchanged and the other images are removed by the brick-wall filter. The interpolation formula always converges absolutely and locally uniformly as long as. This condition is sufficient, but not necessary. Nevertheless, the interpolation formula converges with probability 1.

Convergence can readily be shown by computing the variances of truncated terms of the summation, and showing that the variance can be made arbitrarily small by choosing a sufficient number of terms. If the process mean is nonzero, then pairs of terms need to be considered to also show that the expected value of the truncated terms converges to zero.

Since a random process does not have a Fourier transform, the condition under which the sum converges to the original function must also be different. A stationary random process does have an autocorrelation function and hence a spectral density according to the Wienerâ€”Khinchin theorem. A suitable condition for convergence to a sample function from the process is that the spectral density of the process be zero at all frequencies equal to and above half the sample rate.

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