Generalized Receiver Explanation - Revision history

Jodi.Hodge at 01:32, 9 April 2010

2010-04-09T01:32:30Z

← Older revision		Revision as of 18:32, 8 April 2010
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	<math>y(t)\!</math>		<math>y(t)\!</math>

	The same manipulation is done to get <math>x(t)\!</math> but rather than multiply the input by <math>-2sin(w_{c}t)] \!</math> it is multiplied by <math>2cos(w_{c}t)] \!</math>		The same manipulation is done to get <math>x(t)\!</math> but rather than multiply the input by <math>-2sin(w_{c}t) \!</math> it is multiplied by <math>2cos(w_{c}t) \!</math>

Jodi.Hodge at 01:31, 9 April 2010

2010-04-09T01:31:26Z

← Older revision		Revision as of 18:31, 8 April 2010
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	Theoretically, this how your convert a base band signal to a band pass signal. The band pass signal <math>y(t)\!</math> is the real part of the result of the complex baseband signal <math> g(t)\!</math> multiplied with <math>e^{j2 \pi f_{c}t}\!</math>.		Theoretically, this how your convert a base band signal to a band pass signal. The band pass signal <math>v(t)\!</math> is the real part of the result of the complex baseband signal <math> g(t)\!</math> multiplied with <math>e^{j2 \pi f_{c}t}\!</math>.

	<math> v(t)= Re[g(t)e^{j2 \pi f_{c}t}] \!</math>		<math> v(t)= Re[g(t)e^{j2 \pi f_{c}t}] \!</math>

Jodi.Hodge at 01:29, 9 April 2010

2010-04-09T01:29:18Z

← Older revision		Revision as of 18:29, 8 April 2010
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	Human voice is heard near DC frequencies and is well suited to be packaged in a baseband signal. If a baseband signal is needed to be transmitted, the generalized transmitter will transform the baseband signal into a band pass signal, which has the benefit of requiring only half as much power. After the signal has been transmitted, the job of the generalized receiver is to transform the band pass signal, use for transmitting, back to the original base band signal. Circuit elements needed to create the generalized receiver are an oscillator, two low-pass filters, and two multipliers. The image below a block diagram from Dr. Frohne's Communication Systems class notes.		Human voice is heard near DC frequencies and is well suited to be packaged in a baseband signal. If a baseband signal is needed to be transmitted, the generalized transmitter will transform the baseband signal into a band pass signal, which has the benefit of requiring only half as much power. After the signal has been transmitted, the job of the generalized receiver is to transform the band pass signal, use for transmitting, back to the original base band signal. Circuit elements needed to create the generalized receiver are an oscillator, two low-pass filters, and two multipliers. The image below a block diagram from Dr. Frohne's Communication Systems class notes.

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	'''''Mixers'''''		'''''Mixers'''''

	Lets call the band pass input into the receiver <math>=v(t)\!</math>. Lets call the base band signal <math> g=x(t)+iy(t)\!</math>, where the two outputs of the receiver are its real and imaginary components.		Lets call the band pass input into the receiver <math>v(t)\!</math>. Lets call the base band signal <math> g=x(t)+iy(t)\!</math>, where the two outputs of the receiver are its real and imaginary components.

	Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.		Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.
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	Theoretically, this how your convert a base band signal to a band pass signal. The band pass signal <math>=y(t)\!</math> is the real part of the result of the complex baseband signal <math> g~~=x(t)+iy~~(t)</math> multiplied with <math>e^{j2 \pi f_{c}t}\!</math>.		Theoretically, this how your convert a base band signal to a band pass signal. The band pass signal <math>y(t)\!</math> is the real part of the result of the complex baseband signal <math> g(t)\!</math> multiplied with <math>e^{j2 \pi f_{c}t}\!</math>.

	<math> v(t)= Re[g(t)e^{j2 \pi f_{c}t}]</math>		<math> v(t)= Re[g(t)e^{j2 \pi f_{c}t}] \!</math>

	<math>v(t)=[(x(t)+jy(t))(cos(2 \pi f_{c}t)+jsin(2\pi f_{c}t))]\!</math>		<math>v(t)=[(x(t)+jy(t))(cos(2 \pi f_{c}t)+jsin(2\pi f_{c}t))]\!</math>

Jodi.Hodge at 01:26, 9 April 2010

2010-04-09T01:26:14Z

← Older revision		Revision as of 18:26, 8 April 2010
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	Human voice is heard near DC frequencies and is well suited to be packaged in a baseband signal. If a baseband signal is needed to be transmitted, the generalized transmitter will transform the baseband signal into a band pass signal, which has the benefit of requiring only half as much power. After the signal has been transmitted, the job of the generalized receiver is to transform the band pass signal, use for transmitting, back to the original base band signal. Circuit elements needed to create the generalized receiver are an oscillator, two low-pass filters, and two multipliers. The image below a block diagram from Dr. Frohne's Communication Systems class notes.		Human voice is heard near DC frequencies and is well suited to be packaged in a baseband signal. If a baseband signal is needed to be transmitted, the generalized transmitter will transform the baseband signal into a band pass signal, which has the benefit of requiring only half as much power. After the signal has been transmitted, the job of the generalized receiver is to transform the band pass signal, use for transmitting, back to the original base band signal. Circuit elements needed to create the generalized receiver are an oscillator, two low-pass filters, and two multipliers. The image below a block diagram from Dr. Frohne's Communication Systems class notes.
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	Like what you will do in lab, to create a band pass signal will include the following stages.		Like what you will do in lab, to create a band pass signal will include the following stages.



	''Stage one''		''Stage one''
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	From your oscillator, manipulate the output take two signals; one that is twice the amplitude of the output and another that is twice the amplitude, inverted, and phase shifted by 90 degrees. These signals are <math>2cos(w_{c}t)\!</math> and <math>-2sin(w_{c}t))\!</math> in the block diagram.		From your oscillator, manipulate the output take two signals; one that is twice the amplitude of the output and another that is twice the amplitude, inverted, and phase shifted by 90 degrees. These signals are <math>2cos(w_{c}t)\!</math> and <math>-2sin(w_{c}t))\!</math> in the block diagram.



	''Stage two''		''Stage two''
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	Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.		Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.



	''Stage three''		''Stage three''
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	<math>v(t)=[(x(t)+jy(t))(cos(2 \pi f_{c}t)+jsin(2\pi f_{c}t))]\!</math>		<math>v(t)=[(x(t)+jy(t))(cos(2 \pi f_{c}t)+jsin(2\pi f_{c}t))]\!</math>

	<math>v(t)= x(t)cos(w_{c}t)-y(t)sin(w_{c}t)\!</math		<math>v(t)= x(t)cos(w_{c}t)-y(t)sin(w_{c}t)\!</math>

	<center>[[Image:baseband.jpg]]</center>		<center>[[Image:baseband.jpg]]</center>

	The image shows how the base band signal then the band pass signal look in the frequency domain.		The image shows how the base band signal then the band pass signal look in the frequency domain.

Jodi.Hodge at 01:23, 9 April 2010

2010-04-09T01:23:32Z

← Older revision		Revision as of 18:23, 8 April 2010
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	Like what you will do in lab, to create a band pass signal will include the following stages.		Like what you will do in lab, to create a band pass signal will include the following stages.

	~~'''~~''Stage one''''' '''~~Oscillator~~'''		''Stage one''

			'''''Oscillator'''''

	From your oscillator, manipulate the output take two signals; one that is twice the amplitude of the output and another that is twice the amplitude, inverted, and phase shifted by 90 degrees. These signals are <math>2cos(w_{c}t)\!</math> and <math>-2sin(w_{c}t))\!</math> in the block diagram.		From your oscillator, manipulate the output take two signals; one that is twice the amplitude of the output and another that is twice the amplitude, inverted, and phase shifted by 90 degrees. These signals are <math>2cos(w_{c}t)\!</math> and <math>-2sin(w_{c}t))\!</math> in the block diagram.

	~~'''~~''Stage two''''' '''~~Mixers~~'''		''Stage two''

			'''''Mixers'''''

	Lets call the band pass input into the receiver <math>=v(t)\!</math>. Lets call the base band signal <math> g=x(t)+iy(t)\!</math>, where the two outputs of the receiver are its real and imaginary components.		Lets call the band pass input into the receiver <math>=v(t)\!</math>. Lets call the base band signal <math> g=x(t)+iy(t)\!</math>, where the two outputs of the receiver are its real and imaginary components.

	Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.		Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.

	~~'''~~''Stage three''''' '''Low pass filters'''		''Stage three''

			'''''Low pass filters'''''

	The next stage in your receiver is to send the output of the multipliers into low pass filters (LPF). The LPF bandwidth is near to the frequency of the oscillator which will cut off all other frequencies we do not want.		The next stage in your receiver is to send the output of the multipliers into low pass filters (LPF). The LPF bandwidth is near to the frequency of the oscillator which will cut off all other frequencies we do not want.

Jodi.Hodge at 01:19, 9 April 2010

2010-04-09T01:19:51Z

← Older revision		Revision as of 18:19, 8 April 2010
Line 1:		Line 1:



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	Like what you will do in lab, to create a band pass signal will include the following stages.		Like what you will do in lab, to create a band pass signal will include the following stages.

	Stage one [Oscillator]		'''''Stage one''''' '''Oscillator'''
	From your oscillator, manipulate the output take two signals; one that is twice the amplitude of the output and another that is twice the amplitude, inverted, and phase shifted by 90 degrees. ~~As seen in the picture, this is the~~ <math>2cos(w_{c}t)\!</math> <math>-2sin(w_{c}t))]\!</math> ~~signals~~.		From your oscillator, manipulate the output take two signals; one that is twice the amplitude of the output and another that is twice the amplitude, inverted, and phase shifted by 90 degrees. These signals are <math>2cos(w_{c}t)\!</math> and <math>-2sin(w_{c}t))\!</math> in the block diagram.

	Stage two [Mixers]		'''''Stage two''''' '''Mixers'''
	Lets call the band pass input into the receiver <math>=v(t)\!</math>. Lets call the base band signal <math> g=x(t)+iy(t)\!</math>, where the two outputs of the receiver are its real and imaginary components.		Lets call the band pass input into the receiver <math>=v(t)\!</math>. Lets call the base band signal <math> g=x(t)+iy(t)\!</math>, where the two outputs of the receiver are its real and imaginary components.

	Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.		Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.

	Stage three [Low pass filters]		'''''Stage three''''' '''Low pass filters'''
	The next stage in your receiver is to send the output of the multipliers into low pass filters (LPF). The LPF bandwidth is near to the frequency of the oscillator which will cut off all other frequencies we do not want.		The next stage in your receiver is to send the output of the multipliers into low pass filters (LPF). The LPF bandwidth is near to the frequency of the oscillator which will cut off all other frequencies we do not want.

Jodi.Hodge at 01:13, 9 April 2010

2010-04-09T01:13:01Z

Jodi.Hodge: /* Generalized Receiver */

2010-04-09T01:05:11Z

Generalized Receiver

← Older revision		Revision as of 18:05, 8 April 2010
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	~~== '''Generalized Receiver''' ==~~

	~~----~~


	Human voice is heard near DC frequencies and is well suited to be packaged in a baseband signal. If a baseband signal is needed to be transmitted, the generalized transmitter will transform the baseband signal into a band pass signal, which has the benefit of requiring only half as much power. After the signal has been transmitted, the job of the generalized receiver is to transform the band pass signal, use for transmitting, back to the original base band signal. Circuit elements needed to create the generalized receiver are an oscillator, two low-pass filters, and two multipliers.		Human voice is heard near DC frequencies and is well suited to be packaged in a baseband signal. If a baseband signal is needed to be transmitted, the generalized transmitter will transform the baseband signal into a band pass signal, which has the benefit of requiring only half as much power. After the signal has been transmitted, the job of the generalized receiver is to transform the band pass signal, use for transmitting, back to the original base band signal. Circuit elements needed to create the generalized receiver are an oscillator, two low-pass filters, and two multipliers.

	[[Receiverblock.jpg]]		<center>[[Image:Receiverblock.jpg]]</center>

Jodi.Hodge at 01:02, 9 April 2010

2010-04-09T01:02:43Z

← Older revision		Revision as of 18:02, 8 April 2010
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	~~Generalized Receiver~~

	Human voices are heard near DC frequencies so we want to transform the band pass signal sent from the transmitter back into a baseband signal. The job of the generalized receiver is to transform the band band signal into a base band signal. The circuit elements needed to create the receiver are an oscillator, two low-pass filters, and two multipliers. The benefit of transforming the baseband signal to a band pass is that it requires only half the power to transmit it as band pass signals than as a baseband signal.

	Theoretically, Electronics class notes demonstrates how to create a band pass signal from a baseband signal. Lets called the baseband signal g(t). First, the baseband signal g(t) is multiplied by a function with both sine and cosine with frequency fc, <math>e^{j2 \pi f_{c}t}</math>.The results is the signal in the frequency domain shifted to the location of fc which now looks like G(f-fc). Next we take the real part of this signal, half the conjugate and the normal, is what we now call the band pass signal.		== '''Generalized Receiver''' ==

	~~Experimentally~~, as you will ~~be doing~~ in lab, to create a band pass signal the ~~process is as follows~~. ~~First using~~ your oscillator, ~~you will need to create an original oscillating signal~~ and ~~one~~ that is ~~delayed~~ by 90 degrees, this ~~will be your sine and cosine like~~ the ~~one in~~ the ~~class notes~~. ~~Next~~, ~~you will~~ use your two ~~multipliers~~ to perform the multiplication ~~separately on~~ the ~~baseband~~ signal. The next stage in your receiver is to ~~pass~~ the ~~result~~ of ~~your multiplier~~ into a low pass ~~filter that has~~ bandwidth ~~of frequencies only close~~ to the ~~one~~ of your ~~oscillator~~ to ~~reduce the~~ signal ~~to noise ratio~~. ~~Each~~ signal ~~from~~ the ~~low pass filter is output~~ x(t) ~~and quadrature y~~(t)~~. In the soundcard that your receiver will be connected to will sum both signals together and would match the result we found theoretically.~~		----
	<~~center~~>~~[[Image:baseband.jpg]]~~</~~center~~>

			Human voice is heard near DC frequencies and is well suited to be packaged in a baseband signal. If a baseband signal is needed to be transmitted, the generalized transmitter will transform the baseband signal into a band pass signal, which has the benefit of requiring only half as much power. After the signal has been transmitted, the job of the generalized receiver is to transform the band pass signal, use for transmitting, back to the original base band signal. Circuit elements needed to create the generalized receiver are an oscillator, two low-pass filters, and two multipliers.

			[[Receiverblock.jpg]]


			----

			'''Experimental'''

			Like what you will do in lab, to create a band pass signal will include the following stages.

			Stage one [Oscillator]
			From your oscillator, manipulate the output take two signals; one that is twice the amplitude of the output and another that is twice the amplitude, inverted, and phase shifted by 90 degrees. As seen in the picture, this is the <math>2cos(w_{c}t)\!</math> <math>-2sin(w_{c}t))]\!</math> signals.

			Stage two [Mixers]
			Lets call the band pass input into the receiver <math>=v(t)\!</math>. Lets call the base band signal <math> g=x(t)+iy(t)<math>, where the two outputs of the receiver are its real and imaginary components.

			Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.

			Stage three [Low pass filters]
			The next stage in your receiver is to send the output of the multipliers into low pass filters (LPF). The LPF bandwidth is near to the frequency of the oscillator which will cut off all other frequencies we do not want.

			----


			'''Theoretical'''


			Theoretically, this how your convert a base band signal to a band pass signal. The band pass signal <math>=y(t)\!</math> is the real part of the result of the complex baseband signal <math> g=x(t)+iy(t)<math> multiplied with <math>e^{j2 \pi f_{c}t}</math>.

	<math> v(t)= Re[g(t)e^{j2 \pi f_{c}t}]</math>		<math> v(t)= Re[g(t)e^{j2 \pi f_{c}t}]</math>
			<math>v(t)=[(x(t)+jy(t))(cos(2 \pi f_{c}t)+jsin(2\pi f_{c}t))]\!</math>
			<math>v(t)= x(t)cos(w_{c}t)-y(t)sin(w_{c}t)\!</math
			<center>[[Image:baseband.jpg]]</center>
			The image shows how the base band signal then the band pass signal look in the frequency domain.

	<math>~~v(t)=[(~~x(t)~~+jy(t))(cos(2~~ \~~pi f_{c}t)+jsin~~(~~2\pi f_{c}~~t))]\!</math>		Using cool tricks with of sine and cosine and the low pass filter return<math>x(t)\!</math> and <math>y(t)\!</math> components of the original baseband signal.

			This is the calculation that shows how the band pass signal, the oscillator, and the low pass filter are used.
	<math>-LPF[v(t)2sin(w_{c}t)] \!</math>		<math>-LPF[v(t)2sin(w_{c}t)] \!</math>

			The value of <math>v(t)\!</math> is substituted in

	<math>-LPF[(x(t)cos(w_{c}t)-y(t)sin(w_{c}t)) *2sin(w_{c}t)]\!</math>		<math>-LPF[(x(t)cos(w_{c}t)-y(t)sin(w_{c}t)) *2sin(w_{c}t)]\!</math>

			Multiplication distribution is performed
	<math>-LPF[x(t)cos(w_{c}t)2sin(w_{c}t)-y(t)2sin^{2}(w_{c}t)]\!</math>		<math>-LPF[x(t)cos(w_{c}t)2sin(w_{c}t)-y(t)2sin^{2}(w_{c}t)]\!</math>

	~~using trigonometry~~ identities		Trigonometry identities are used
	<math>-LPF[x(t)sin(2w_{c}t)-y(t)(1-cos(w_{c}t)]\!</math>		<math>-LPF[x(t)sin(2w_{c}t)-y(t)(1-cos(w_{c}t)]\!</math>

	~~Implementing the Lowpass~~ filter to remove frequencies above wc ~~will remove a number of~~ terms		The low pass filter is used to remove frequencies above wc, in this case all the terms with 2wc
	<math>-LPF[x(t)sin(2w_{c}t)-y(t)(1-cos(w_{c}t)]\!</math>		<math>-LPF[x(t)sin(2w_{c}t)-y(t)(1-cos(2w_{c}t)]\!</math>
	<math>=y(t)\!</math>
			The only signal that remains after the low pass filter is
			<math>y(t)\!</math>

			The same manipulation is done to get <math>x(t)\!</math> but rather than multiply the input by <math>-2sin(w_{c}t)] \!</math> it is multiplied by<math>2cos(w_{c}t)] \!</math>

Jodi.Hodge at 07:01, 6 April 2010

2010-04-06T07:01:39Z

← Older revision		Revision as of 00:01, 6 April 2010
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	Generalized Receiver		Generalized Receiver

	Human voices are heard near DC frequencies ~~and conviently~~ the ~~input~~ signal from the ~~antenna is~~ a baseband signal. The job of the generalized receiver is to transform the ~~base~~ band signal into a base band signal ~~that has~~ an ~~upper~~ and ~~lower part~~. The benefit of transforming the baseband signal to a band pass is that it requires only half the power to transmit it as band pass signals ~~rather~~ than a baseband signal.~~The circuit elements needed to create the receiver are an oscillator, two low-pass filters, and two multipliers.~~		Human voices are heard near DC frequencies so we want to transform the band pass signal sent from the transmitter back into a baseband signal. The job of the generalized receiver is to transform the band band signal into a base band signal. The circuit elements needed to create the receiver are an oscillator, two low-pass filters, and two multipliers. The benefit of transforming the baseband signal to a band pass is that it requires only half the power to transmit it as band pass signals than as a baseband signal.

	Theoretically, Electronics class notes demonstrates how to create a band pass signal from a baseband signal. Lets called the baseband signal g(t). First, the baseband signal g(t) is multiplied by a function with both sine and cosine with frequency fc, <math>e^{j2 \pi f_{c}t}</math>.The results is the signal in the frequency domain shifted to the location of fc which now looks like G(f-fc). Next we take the real part of this signal, half the conjugate and the normal, is what we now call the band pass signal.		Theoretically, Electronics class notes demonstrates how to create a band pass signal from a baseband signal. Lets called the baseband signal g(t). First, the baseband signal g(t) is multiplied by a function with both sine and cosine with frequency fc, <math>e^{j2 \pi f_{c}t}</math>.The results is the signal in the frequency domain shifted to the location of fc which now looks like G(f-fc). Next we take the real part of this signal, half the conjugate and the normal, is what we now call the band pass signal.

← Older revision		Revision as of 18:13, 8 April 2010
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			Human voice is heard near DC frequencies and is well suited to be packaged in a baseband signal. If a baseband signal is needed to be transmitted, the generalized transmitter will transform the baseband signal into a band pass signal, which has the benefit of requiring only half as much power. After the signal has been transmitted, the job of the generalized receiver is to transform the band pass signal, use for transmitting, back to the original base band signal. Circuit elements needed to create the generalized receiver are an oscillator, two low-pass filters, and two multipliers. The image below a block diagram from Dr. Frohne's Communication Systems class notes.
	Human voice is heard near DC frequencies and is well suited to be packaged in a baseband signal. If a baseband signal is needed to be transmitted, the generalized transmitter will transform the baseband signal into a band pass signal, which has the benefit of requiring only half as much power. After the signal has been transmitted, the job of the generalized receiver is to transform the band pass signal, use for transmitting, back to the original base band signal. Circuit elements needed to create the generalized receiver are an oscillator, two low-pass filters, and two multipliers.

	<center>[[Image:Receiverblock.jpg]]</center>		<center>[[Image:Receiverblock.jpg]]</center>
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	Stage two [Mixers]		Stage two [Mixers]
	Lets call the band pass input into the receiver <math>=v(t)\!</math>. Lets call the base band signal <math> g=x(t)+iy(t)<math>, where the two outputs of the receiver are its real and imaginary components.		Lets call the band pass input into the receiver <math>=v(t)\!</math>. Lets call the base band signal <math> g=x(t)+iy(t)\!</math>, where the two outputs of the receiver are its real and imaginary components.

	Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.		Mixers work as multipliers. So the next step in building your general receiver is to use your mixers to perform multiplication of your input with your oscillator. Since we need two separate outputs, let’s have one mixer perform the multiplication of the band pass signal with cosine and the other mixer to perform the multiplication of the band pass signal with negative sine.
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	Theoretically, this how your convert a base band signal to a band pass signal. The band pass signal <math>=y(t)\!</math> is the real part of the result of the complex baseband signal <math> g=x(t)+iy(t)<math> multiplied with <math>e^{j2 \pi f_{c}t}</math>.		Theoretically, this how your convert a base band signal to a band pass signal. The band pass signal <math>=y(t)\!</math> is the real part of the result of the complex baseband signal <math> g=x(t)+iy(t)</math> multiplied with <math>e^{j2 \pi f_{c}t}\!</math>.

	<math> v(t)= Re[g(t)e^{j2 \pi f_{c}t}]</math>		<math> v(t)= Re[g(t)e^{j2 \pi f_{c}t}]</math>

	<math>v(t)=[(x(t)+jy(t))(cos(2 \pi f_{c}t)+jsin(2\pi f_{c}t))]\!</math>		<math>v(t)=[(x(t)+jy(t))(cos(2 \pi f_{c}t)+jsin(2\pi f_{c}t))]\!</math>

	<math>v(t)= x(t)cos(w_{c}t)-y(t)sin(w_{c}t)\!</math		<math>v(t)= x(t)cos(w_{c}t)-y(t)sin(w_{c}t)\!</math

	<center>[[Image:baseband.jpg]]</center>		<center>[[Image:baseband.jpg]]</center>
	The image shows how the base band signal then the band pass signal look in the frequency domain.		The image shows how the base band signal then the band pass signal look in the frequency domain.

	Using cool tricks with of sine and cosine and the low pass filter return<math>x(t)\!</math> and <math>y(t)\!</math> components of the original baseband signal.		Using cool tricks with of sine and cosine and the low pass filter return<math>x(t)\!</math> and <math>y(t)\!</math> components of the original baseband signal.

	This is the calculation that shows how the band pass signal, the oscillator, and the low pass filter are used.		This is the calculation that shows how the band pass signal, the oscillator, and the low pass filter are used.

	<math>-LPF[v(t)2sin(w_{c}t)] \!</math>		<math>-LPF[v(t)2sin(w_{c}t)] \!</math>

Line 50:		Line 53:

	Multiplication distribution is performed		Multiplication distribution is performed

	<math>-LPF[x(t)cos(w_{c}t)2sin(w_{c}t)-y(t)2sin^{2}(w_{c}t)]\!</math>		<math>-LPF[x(t)cos(w_{c}t)2sin(w_{c}t)-y(t)2sin^{2}(w_{c}t)]\!</math>

	Trigonometry identities are used		Trigonometry identities are used

	<math>-LPF[x(t)sin(2w_{c}t)-y(t)(1-cos(w_{c}t)]\!</math>		<math>-LPF[x(t)sin(2w_{c}t)-y(t)(1-cos(w_{c}t)]\!</math>

	The low pass filter is used to remove frequencies above wc, in this case all the terms with 2wc		The low pass filter is used to remove frequencies above wc, in this case all the terms with 2wc

	<math>-LPF[x(t)sin(2w_{c}t)-y(t)(1-cos(2w_{c}t)]\!</math>		<math>-LPF[x(t)sin(2w_{c}t)-y(t)(1-cos(2w_{c}t)]\!</math>

	The only signal that remains after the low pass filter is		The only signal that remains after the low pass filter is

	<math>y(t)\!</math>		<math>y(t)\!</math>

	The same manipulation is done to get <math>x(t)\!</math> but rather than multiply the input by <math>-2sin(w_{c}t)] \!</math> it is multiplied by<math>2cos(w_{c}t)] \!</math>		The same manipulation is done to get <math>x(t)\!</math> but rather than multiply the input by <math>-2sin(w_{c}t)] \!</math> it is multiplied by <math>2cos(w_{c}t)] \!</math>