Tower Mounted Amplifier - Revision history

en>Hooperbloob: copyedit

2014-07-03T15:32:43Z

copyedit

← Older revision		Revision as of 17:32, 3 July 2014
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en>Smtchahal: Reverted edits by 135.245.115.245 (talk) to last revision by Al Doerksen (HG)

2014-02-07T08:08:51Z

Reverted edits by 135.245.115.245 (talk) to last revision by Al Doerksen (HG)

← Older revision		Revision as of 10:08, 7 February 2014
Line 1:		Line 1:
	~~{{More footnotes\|date=April 2009}}~~		The name of the writer is Jayson. Some time in the past she chose to reside in Alaska and her parents live close by. Office supervising is where my primary income arrives from but I've usually needed my own business. The favorite pastime for him and his children is style and he'll be beginning something else along with it.<br><br>Look at my web blog; best psychic ([http://test.jeka-nn.ru/node/129 http://test.jeka-nn.ru])

	'''Impulse invariance''' is a technique for designing discrete-time [[infinite-impulse-response]] (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discrete-time system. The ~~frequency response of the discrete-time system will be a sum of shifted copies of the frequency response~~ of the ~~continuous-time system; if the continuous-time system~~ is approximately band-limited to a frequency less than the [[Nyquist frequency]] of the sampling, then the frequency response of the discrete-time system will be approximately equal to it for frequencies below the Nyquist frequency.

	~~==Discussion==~~

	~~The continuous-~~time ~~system's impulse response, <math>h_c(t)</math>, is sampled with sampling period <math>T</math> to produce the discrete-time system's impulse response, <math>h[n]</math>.~~

	~~:<math>h[n]=Th_c(nT)\,</math>~~

	~~Thus, the frequency responses of the two systems are related by~~

	~~:<math>H(e^{j\omega}) = \sum_{k=-\infty}^\infty{H_c\left(j\frac{\omega}{T} + j\frac{2{\pi}}{T}k\right)}\,</math>~~

	If the continuous time filter is approximately band-limited (i.e. <math>H_c(j\Omega) < \delta</math> when <math>\|\Omega\| \ge \pi/T</math>), then the frequency response of the discrete-time system will be approximately the continuous-time system's frequency response for frequencies below π radians per sample (below the Nyquist frequency 1/(2''T'') Hz):

	~~:<math>H(e^{j\omega}) = H_c(j\omega/T)\,</math> for <math>\|\omega\| \le \pi\,</math>~~

	~~===Comparison to~~ the ~~bilinear transform===~~

	~~Note that aliasing will occur, including aliasing below the Nyquist frequency~~ to the extent that the continuous-time filter's response is nonzero above that frequency. The [[bilinear transform]] is an alternative to impulse invariance that uses a different mapping that maps the continuous-time system's frequency response, out to infinite frequency, into the range of frequencies up to the Nyquist frequency in ~~the discrete-time case, as opposed to mapping frequencies linearly with circular overlap as impulse invariance does~~.

	~~===Effect on poles in system function===~~

	~~If the continuous poles at <math>s = s_k</math>, the system function can be written in partial fraction expansion as~~

	~~:<math>H_c(s) = \sum_{k=1}^N{\frac{A_k}{s-s_k}}\,</math>~~

	~~Thus, using the inverse Laplace transform, the impulse response~~ is

	~~:<math>h_c(t) = \begin{cases}~~
	~~\sum_{k=1}^N{A_ke^{s_kt}}, & t \ge 0 \\~~
	~~0, & \mbox{otherwise}~~
	~~\end{cases}</math>~~

	~~The corresponding discrete-time system~~'~~s impulse response is then defined as the following~~

	~~:<math>h[n] = Th_c(nT)\,</math>~~

	~~:<math>h[n] = T \sum_{k=1}^N{A_ke^{s_knT}u[n]}\,</math>~~

	~~Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function~~

	~~:<math>H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}}}\,</math>~~

	~~Thus the poles from the continuous-time system function are translated to poles at z = e<sup>s<sub>k</sub>T</sup>~~. The ~~zeros, if any, are not so simply mapped.{{clarify\|date=February 2013}}~~

	~~===Poles~~ and ~~zeros===~~

	If the system function has zeros as well as poles, they can be mapped the same way, but the result is no longer an impulse invariance result: the discrete-time impulse response is not equal simply to samples of the continuous-time impulse response. This method is ~~known as the [[matched Z-transform method]], or pole–zero mapping. In the case of all-pole filters, the methods are equivalent.~~

	~~===Stability~~ and ~~causality===~~

	~~Since poles in the continuous-time system at~~ '~~'s'' = ''s~~<~~sub~~>k<~~/sub~~>~~'' transform to poles in the discrete-time system~~ at ~~z = exp~~(''s<sub>k</sub>T''), poles in the left half of the ''s''-plane map to inside the unit circle in the ''z''-plane; so if the continuous-time filter is causal and stable, then the discrete-time filter will be causal and stable as well.

	~~===Corrected formula===~~

	~~When a causal continuous-time impulse response has a discontinuity at <math>t=0</math>, the expressions above are not consistent.<ref>~~[http://~~ieeexplore~~.~~ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=870677&isnumber=18861] L. Jackson, "A correction to impulse invariance", IEEE Signal Processing Letters, Vol. 7, Oct~~. ~~2000.</ref>~~
	~~This is because <math>h_c (0)<~~/~~math> should really only contribute half its value to <math>h[0]<~~/~~math>.~~

	~~Making this correction gives~~

	~~:<math>h[n] = T \left( h_c(nT) - \frac{1}{2} h_c(0)\delta [n] \right) \,</math>~~

	~~:<math>h[n] = T \sum_{k=1}^N{A_ke^{s_knT}} \left( u[n] - \frac{1}{2} \delta[n] \right) \,</math>~~

	~~Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function~~

	~~:<math>H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}} - \frac{T}{2} \sum_{k=1}^N A_k}.</math>~~

	~~==See also==~~

	* [[Infinite impulse response]]
	* [[Bilinear transform]]
	* [[Matched Z-transform method]]
	* Continuous-time filters:
	*:[[Chebyshev filter]]
	*:[[Butterworth filter]]
	*:[[Elliptic filter]]

	~~==References==~~

	~~{{Reflist}}~~

	~~===Other sources===~~

	~~{{Refbegin}}~~
	* Oppenheim, Alan V. and Schafer, Ronald W. with Buck, John R. ''Discrete-Time Signal Processing.'' Second Edition. Upper Saddle River, New Jersey: Prentice-Hall, 1999.
	* Sahai, Anant. Course Lecture. Electrical Engineering 123: Digital Signal Processing. University of California, Berkeley. 5 April 2007.

	~~==External links==~~
	* [http://~~www.circuitdesign~~.~~info/blog/2009/02/impulse~~-~~invariant-transform/ Impulse Invariant Transform at CircuitDesign~~.~~info] Brief explanation, an example, and application to [[Continuous signal\|Continuous Time]] [[Sigma Delta]] ADC's.~~
	~~{{Refend}}~~

	~~<!-- AWB bots, leave this extra space alone! -->~~

	~~{{DSP}}~~
	~~{{Use dmy dates\|date=September 2010}}~~

	~~{{DEFAULTSORT:Impulse Invariance}}~~
	~~[[Category:Digital signal processing]]~~
	~~[[Category:Filter theory]~~]

en>Al Doerksen at 23:32, 4 March 2013

2013-03-04T23:32:30Z

← Older revision		Revision as of 01:32, 5 March 2013
Line 1:		Line 1:
	~~Andrew Simcox~~ is the ~~name his parents gave him and he completely enjoys this title~~. ~~Ohio~~ is ~~where my home~~ is ~~but my spouse wants us~~ to ~~transfer~~. ~~To perform domino~~ is ~~something I truly appreciate performing~~. ~~Credit authorising~~ is ~~where my primary income comes~~ from.<br><br>~~My blog tarot readings~~ ([http://~~www~~.~~zavodpm~~.ru/~~blogs~~/~~glennmusserrvji~~/~~14565~~-~~great~~-~~hobby~~-~~advice~~-~~assist~~-~~allow~~-~~you~~-~~get~~-~~going click the up coming website~~])		{{More footnotes\|date=April 2009}}

			'''Impulse invariance''' is a technique for designing discrete-time [[infinite-impulse-response]] (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discrete-time system. The frequency response of the discrete-time system will be a sum of shifted copies of the frequency response of the continuous-time system; if the continuous-time system is approximately band-limited to a frequency less than the [[Nyquist frequency]] of the sampling, then the frequency response of the discrete-time system will be approximately equal to it for frequencies below the Nyquist frequency.

			==Discussion==

			The continuous-time system's impulse response, <math>h_c(t)</math>, is sampled with sampling period <math>T</math> to produce the discrete-time system's impulse response, <math>h[n]</math>.

			:<math>h[n]=Th_c(nT)\,</math>

			Thus, the frequency responses of the two systems are related by

			:<math>H(e^{j\omega}) = \sum_{k=-\infty}^\infty{H_c\left(j\frac{\omega}{T} + j\frac{2{\pi}}{T}k\right)}\,</math>

			If the continuous time filter is approximately band-limited (i.e. <math>H_c(j\Omega) < \delta</math> when <math>\|\Omega\| \ge \pi/T</math>), then the frequency response of the discrete-time system will be approximately the continuous-time system's frequency response for frequencies below π radians per sample (below the Nyquist frequency 1/(2''T'') Hz):

			:<math>H(e^{j\omega}) = H_c(j\omega/T)\,</math> for <math>\|\omega\| \le \pi\,</math>

			===Comparison to the bilinear transform===

			Note that aliasing will occur, including aliasing below the Nyquist frequency to the extent that the continuous-time filter's response is nonzero above that frequency. The [[bilinear transform]] is an alternative to impulse invariance that uses a different mapping that maps the continuous-time system's frequency response, out to infinite frequency, into the range of frequencies up to the Nyquist frequency in the discrete-time case, as opposed to mapping frequencies linearly with circular overlap as impulse invariance does.

			===Effect on poles in system function===

			If the continuous poles at <math>s = s_k</math>, the system function can be written in partial fraction expansion as

			:<math>H_c(s) = \sum_{k=1}^N{\frac{A_k}{s-s_k}}\,</math>

			Thus, using the inverse Laplace transform, the impulse response is

			:<math>h_c(t) = \begin{cases}
			\sum_{k=1}^N{A_ke^{s_kt}}, & t \ge 0 \\
			0, & \mbox{otherwise}
			\end{cases}</math>

			The corresponding discrete-time system's impulse response is then defined as the following

			:<math>h[n] = Th_c(nT)\,</math>

			:<math>h[n] = T \sum_{k=1}^N{A_ke^{s_knT}u[n]}\,</math>

			Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function

			:<math>H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}}}\,</math>

			Thus the poles from the continuous-time system function are translated to poles at z = e<sup>s<sub>k</sub>T</sup>. The zeros, if any, are not so simply mapped.{{clarify\|date=February 2013}}

			===Poles and zeros===

			If the system function has zeros as well as poles, they can be mapped the same way, but the result is no longer an impulse invariance result: the discrete-time impulse response is not equal simply to samples of the continuous-time impulse response. This method is known as the [[matched Z-transform method]], or pole–zero mapping. In the case of all-pole filters, the methods are equivalent.

			===Stability and causality===

			Since poles in the continuous-time system at ''s'' = ''s<sub>k</sub>'' transform to poles in the discrete-time system at z = exp(''s<sub>k</sub>T''), poles in the left half of the ''s''-plane map to inside the unit circle in the ''z''-plane; so if the continuous-time filter is causal and stable, then the discrete-time filter will be causal and stable as well.

			===Corrected formula===

			When a causal continuous-time impulse response has a discontinuity at <math>t=0</math>, the expressions above are not consistent.<ref>[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=870677&isnumber=18861] L. Jackson, "A correction to impulse invariance", IEEE Signal Processing Letters, Vol. 7, Oct. 2000.</ref>
			This is because <math>h_c (0)</math> should really only contribute half its value to <math>h[0]</math>.

			Making this correction gives

			:<math>h[n] = T \left( h_c(nT) - \frac{1}{2} h_c(0)\delta [n] \right) \,</math>

			:<math>h[n] = T \sum_{k=1}^N{A_ke^{s_knT}} \left( u[n] - \frac{1}{2} \delta[n] \right) \,</math>

			Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function

			:<math>H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}} - \frac{T}{2} \sum_{k=1}^N A_k}.</math>

			==See also==

			* [[Infinite impulse response]]
			* [[Bilinear transform]]
			* [[Matched Z-transform method]]
			* Continuous-time filters:
			*:[[Chebyshev filter]]
			*:[[Butterworth filter]]
			*:[[Elliptic filter]]

			==References==

			{{Reflist}}

			===Other sources===

			{{Refbegin}}
			* Oppenheim, Alan V. and Schafer, Ronald W. with Buck, John R. ''Discrete-Time Signal Processing.'' Second Edition. Upper Saddle River, New Jersey: Prentice-Hall, 1999.
			* Sahai, Anant. Course Lecture. Electrical Engineering 123: Digital Signal Processing. University of California, Berkeley. 5 April 2007.

			==External links==
			* [http://www.circuitdesign.info/blog/2009/02/impulse-invariant-transform/ Impulse Invariant Transform at CircuitDesign.info] Brief explanation, an example, and application to [[Continuous signal\|Continuous Time]] [[Sigma Delta]] ADC's.
			{{Refend}}

			<!-- AWB bots, leave this extra space alone! -->

			{{DSP}}
			{{Use dmy dates\|date=September 2010}}

			{{DEFAULTSORT:Impulse Invariance}}
			[[Category:Digital signal processing]]
			[[Category:Filter theory]]

en>ABACA: /* Drawbacks/pitfalls */ broken list repaired

2012-03-29T17:06:12Z

Drawbacks/pitfalls: broken list repaired

New page

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