Class Wiki - User contributions [en]

User:Andeda

2004-12-10T23:59:21Z

Andeda:

[[Image:David.jpg|thumb|David's photo]]

== Welcome to David's Wiki Page! ==

Click on the link below to go to my page describing how a CD Player works.

[[DavidsCD]]

The following links go to things I've worked on to get hours for HW#13. I accumulated 3 hours total.

[[FIR_Filters]]

[[Aliasing]]

Aliasing

2004-12-10T23:58:33Z

Andeda:

Alaising is what occurs when you sample the highest frequency component of a signal less than two times and then are not able to reconstruct the original signal from the sampled data. This is generalized by Nyquists sampling theorem that says the signal must be sampled at a frequency of 1/T > 2*fmax. In the example shown below a wave has a sample period of T and thus a sampling frequency of 1/T. fmax in this case is greater than 1/2T so there is some overlap in the bandwidth. This overlap of X(f) will add constructively in the sampled signal. Now when we wish to convert this sampled signal back to the the original we can not because we have no idea what it orginally looked like, or what frequency components it had.

[[Image:alias.gif]]

Aliasing

2004-12-10T23:56:46Z

Andeda:

Aliasing

2004-12-10T23:55:43Z

Andeda:

Alaising is what occurs when you sample the highest frequency component of a signal less than two times and then are not able to reconstruct the original signal from the sampled data. This is generalized by Nyquists sampling theorem that says the signal must be sampled at a frequency of 1/T > 2*fmax. In the example shown below a wave has a sample period of T and thus a sampling frequency of 1/T. fmax in this case is greater than 1/2T so there is some overlap in the bandwidth. This overlap of X(f) will add constructively in the sampled. Now when we wish to convert this sampled signal back to the the original we can not because we have no idea what it orginally looked like.

[[Image:alias.gif]]

Aliasing

2004-12-10T23:46:08Z

Andeda:

File:Alias.gif

2004-12-10T23:45:47Z

Andeda:

Aliasing

2004-12-10T23:22:05Z

Andeda:

Aliasing

2004-12-10T22:56:38Z

Andeda:

Alaising is what occurs when you are sampling to slow in the time domain.

User:Andeda

2004-12-10T22:49:29Z

Andeda: /* Welcome to David's Wiki Page! */

User:Andeda

2004-12-10T22:48:44Z

Andeda: /* Welcome to David's Wiki Page! */

[[Image:David.jpg|thumb|David's photo]]

== Welcome to David's Wiki Page! ==

Click on the link below to go to my page describing how a CD Player works.

[[DavidsCD]]

The following link goes to the FIR section I worked on

[[FIR_Filters]]

[[Aliasing]]

FIR Filters

2004-12-10T22:37:04Z

Andeda: /* Finite Impulse Response Filters */

== Finite Impulse Response Filters ==
[[Image:firtap.jpg]]

Above is shown the Tap Delay Line version of an FIR filter. It can actually be created in an analog circuit using transmission lines for the T Delay, resistors, an opamp, and an inverting summer to sum them all together. In the digital sense whats really happening here is that the output of the FIR filter (y(l)) is the convolution of h(l) with x(l) as shown below.

[[Image:firgarbage.jpg]]

This filter can simply be viewed as a weighted average of the value around the time you are currently at, where h(l) would be the weighting coefficients. Mathematically it can be viewed as a multiply and accumulate because we are simply taking a predefined impulse function h() and multiplying it by x() then taking the next value of h() and multiplying it by the next value of x(). Then you just sum all those value together and that is the value of some y() at a certain time. Then you would move forward a step in time and throw out your oldest x() value and bring in the newest x() value to the top of your stack as shown in the picture.

FIR Filters

2004-12-10T12:33:01Z

Andeda:

== Finite Impulse Response Filters ==
[[Image:firtap.jpg]]

Above is shown the Tap Delay Line version of an FIR filter. It can actually be created in an analog circuit using transmission lines for the T Delay, resistors, an opamp, and an inverting summer to sum them all together. In the digital sense whats really happening here is that the output of the FIR filter (y(l)) is the convolution of h(l) with x(l) as shown below.

[[Image:firgarbage.jpg]]

This filter can simply be viewed as a weighted average of the value around the time you are currently at, where h(l) would be the weighting coefficients. Mathmatically it can be viewed as a multiply and accumulate becuase we are simply taking a predifined impulse function h() and multiplying it by x() then taking the next value of h() and multiplying it by the next value of x(). Then you just sum all those value together and that is the value of some y() at a certain time. Then you would move forward a step in time and throw out your oldest x() value and bring in the newest x() value to the top of your stack as shown in the picture.

FIR Filters

2004-12-10T12:29:39Z

Andeda:

== Finite Impulse Response Filters ==
[[Image:firtap.jpg]]

Above is shown the Tap Delay Line version of an FIR filter. It can actually be created in an analog circuit using transmission lines for the T Delay, resistors, an opamp, and an inverting summer to sum them all together. In the digital sense whats really happening here is that the output of the FIR filter (y(l)) is the convolution of h(l) with x(l) as shown below.

[[Image:firgarbage.jpg]]

This filter can simply be viewed as a weighted average of the value around the time you are currently at, where h(l) would be the weighting coefficients. Mathmatically it can be viewed as a multiply and accumulate becuase we are simply taking a predifined impulse function h() and multiplying it by x() then taking the next value of h() and multiplying it by the next value of x(). Then you just sum all those value together and that is the value of some y() at a certain time.

FIR Filters

2004-12-10T12:22:19Z

Andeda: /* Finite Impulse Response Filters */

FIR Filters

2004-12-10T12:14:20Z

Andeda:

File:Firgarbage.jpg

2004-12-10T11:53:54Z

Andeda:

FIR Filters

2004-12-10T11:53:39Z

Andeda:

== Finite Impulse Response Filters ==
[[Image:firtap.jpg]]

The output of an FIR filter (y(l)) is the convolution of h(l) with x(l). It can simply be viewed as a multiply and accumulate.

[[Image:firgarbage.jpg]]

FIR Filters

2004-12-10T11:47:15Z

Andeda:

== Finite Impulse Response Filters ==
[[Image:firtap.jpg]]

The output of an FIR filter (y(l)) is the convolution of h(l) with x(l). It can simply be viewed as a multiply and accumulate.

FIR Filters

2004-12-10T11:28:52Z

Andeda: /* Finite Impulse Response Filters */

== Finite Impulse Response Filters ==
[[Image:firtap.jpg]]

File:Firtap.jpg

2004-12-10T11:28:12Z

Andeda:

FIR Filters

2004-12-10T11:06:49Z

Andeda:

== Finite Impulse Response Filters ==

User:Andeda

2004-12-10T11:06:15Z

Andeda: /* Welcome to David's Wiki Page! */

[[Image:David.jpg|thumb|David's photo]]

== Welcome to David's Wiki Page! ==

Click on the link below to go to my page describing how a CD Player works.

[[DavidsCD]]

The following link goes to the FIR section I worked on

[[FIR_Filters]]

User:Andeda

2004-12-10T11:04:35Z

Andeda: /* Welcome to David's Wiki Page! */

Signals and Systems

2004-12-10T11:00:28Z

Andeda: /* Topics */

[http://www.wwc.edu/~frohro/ClassNotes/engr455index.htm Class notes for Signals & Systems]

== Topics ==
*[[Orthogonal functions]]
*[[Fourier series]]
*[[Fourier transform]]
*[[Sampling]]
*[[Discrete Fourier transform]]

I couldn't figure out how to get to others Users pages easily so I decided to start posting them here, please add yours:

[[User:Frohro|Rob Frohne]]

[[User:Barnsa|Sam Barnes]]

[[User:Santsh|Shawn Santana]]

[[User:Goeari|Aric Goe]]

[[User:Caswto|Todd Caswell]]

[[User:Andeda|David Anderson]]

File:Davespk.gif

2004-12-10T10:56:23Z

Andeda:

DavidsCD

2004-12-10T10:53:58Z

Andeda:

== CD Players Explained! ==

[[Image:davemic.gif]]

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

[[Image:davespk.gif]]

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC) it is convolved with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math> as shown below.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

In the frequency domain you can see that this relates to multipliing <math> \frac{1}{T} \sum_{n=-\infty}^\infty X(f - \frac{n}{T}) \cdot </math> by P(f) and results in a quite distored X(f). It has to many high frequency components and would require a really good brick wall filter to get rid of them. From there the signal is sent to a Low Pass Filter where the stair stepped shaped function is smoothed so that it sounds better when the signal is next sent to the speaker.

== Two Times Oversampling ==

Oversampling is the process of interpolating data so that it looks like we have more data than we really do. Two times oversampling is accomplished by adding a digital interpolation filter right before the DAC. Now <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math> is convolved with the desired impulse response. For two times oversampling it would be convolved with <math> h(lt/2) = \frac{T}{2} \int_{-\frac{-1}{T}}^{\frac{1}{T}} e^{(j 2 \pi l t f)/2} df </math> if we wanted to predistort the signal as well. This makes it so that the resulting wave in the frequency domain more closely matches the original signal. So we get more data points doing it this way, we get to filter the signal however we want, plus this allows for the use of a cheaper low pass filter since the frequency spacing is now 2/T. We just have to be sure and meet the Nyquist criteria now and sample the DAC at T/2 or 88kHz.

File:Davemic.gif

2004-12-10T10:53:13Z

Andeda:

DavidsCD

2004-12-10T10:41:11Z

Andeda: /* Two Times Oversampling */

== CD Players Explained! ==

insert pic1

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

insert pic2

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC) it is convolved with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math> as shown below.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

In the frequency domain you can see that this relates to multipliing <math> \frac{1}{T} \sum_{n=-\infty}^\infty X(f - \frac{n}{T}) \cdot </math> by P(f) and results in a quite distored X(f). It has to many high frequency components and would require a really good brick wall filter to get rid of them. From there the signal is sent to a Low Pass Filter where the stair stepped shaped function is smoothed so that it sounds better when the signal is next sent to the speaker.

== Two Times Oversampling ==

Oversampling is the process of interpolating data so that it looks like we have more data than we really do. Two times oversampling is accomplished by adding a digital interpolation filter right before the DAC. Now <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math> is convolved with the desired impulse response. For two times oversampling it would be convolved with <math> h(lt/2) = \frac{T}{2} \int_{-\frac{-1}{T}}^{\frac{1}{T}} e^{(j 2 \pi l t f)/2} df </math> if we wanted to predistort the signal as well. This makes it so that the resulting wave in the frequency domain more closely matches the original signal. So we get more data points doing it this way, we get to filter the signal however we want, plus this allows for the use of a cheaper low pass filter since the frequency spacing is now 2/T. We just have to be sure and meet the Nyquist criteria now and sample the DAC at T/2 or 88kHz.

DavidsCD

2004-12-10T10:37:26Z

Andeda: /* CD Players Explained! */

DavidsCD

2004-12-10T10:37:13Z

Andeda: /* Two Times Oversampling */

= CD Players Explained! =

insert pic1

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

insert pic2

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC) it is convolved with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math> as shown below.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

In the frequency domain you can see that this relates to multipliing <math> \frac{1}{T} \sum_{n=-\infty}^\infty X(f - \frac{n}{T}) \cdot </math> by P(f) and results in a quite distored X(f). It has to many high frequency components and would require a really good brick wall filter to get rid of them. From there the signal is sent to a Low Pass Filter where the stair stepped shaped function is smoothed so that it sounds better when the signal is next sent to the speaker.

== Two Times Oversampling ==

Oversampling is the process of interpolating data so that it looks like we have more data than we really do. Two times oversampling is accomplished by adding a digital interpolation filter right before the DAC. Now <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math> is convolved with the desired impulse response. For two times oversampling it would be convolved with <math> h(lt/2) = \frac{T}{2} \int_{-\frac{-1}{T}}^{\frac{1}{T}} e^{(j 2 \pi l t f)/2} df </math> if we wanted to predistort the signal as well. This makes it so that the resulting wave in the frequency domain more closely matches the original signal.

DavidsCD

2004-12-10T10:32:59Z

Andeda: /* Two Times Oversampling */

= CD Players Explained! =

insert pic1

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

insert pic2

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC) it is convolved with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math> as shown below.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

In the frequency domain you can see that this relates to multipliing <math> \frac{1}{T} \sum_{n=-\infty}^\infty X(f - \frac{n}{T}) \cdot </math> by P(f) and results in a quite distored X(f). It has to many high frequency components and would require a really good brick wall filter to get rid of them. From there the signal is sent to a Low Pass Filter where the stair stepped shaped function is smoothed so that it sounds better when the signal is next sent to the speaker.

= Two Times Oversampling =

Oversampling is the process of interpolating data so that it looks like we have more data than we really do. Two times oversampling is accomplished by adding a digital interpolation filter right before the DAC. Now <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math> is convolved with the desired impulse response. For two times oversampling it would be convolved with <math> h(lt/2) = \frac{T}{2} \int_{-\frac{-1}{T}}^{\frac{1}{T}} e^{(j 2 \pi l t f)/2} df </math>.

DavidsCD

2004-12-10T10:24:44Z

Andeda: /* Two Times Oversampling */

= CD Players Explained! =

insert pic1

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

insert pic2

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC) it is convolved with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math> as shown below.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

In the frequency domain you can see that this relates to multipliing <math> \frac{1}{T} \sum_{n=-\infty}^\infty X(f - \frac{n}{T}) \cdot </math> by P(f) and results in a quite distored X(f). It has to many high frequency components and would require a really good brick wall filter to get rid of them. From there the signal is sent to a Low Pass Filter where the stair stepped shaped function is smoothed so that it sounds better when the signal is next sent to the speaker.

= Two Times Oversampling =

Oversampling is the process of interpolating data so that it looks like we have more data than we really do. Two times oversampling is accomplished by adding a digital interpolation filter right before the DAC. Now <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math> is convolved with h(t) the desired impulse response. For two times oversampling h(t) would be by <math>h(lt/2) = frac{T}{2}\int_{-\frac{-1}{T}}^{\frac{1}{T}} e^{j*2*pi*l*t*f\2} \, df </math>

DavidsCD

2004-12-10T10:17:44Z

Andeda:

= CD Players Explained! =

insert pic1

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

insert pic2

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC) it is convolved with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math> as shown below.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

In the frequency domain you can see that this relates to multipliing <math> \frac{1}{T} \sum_{n=-\infty}^\infty X(f - \frac{n}{T}) \cdot </math> by P(f) and results in a quite distored X(f). It has to many high frequency components and would require a really good brick wall filter to get rid of them. From there the signal is sent to a Low Pass Filter where the stair stepped shaped function is smoothed so that it sounds better when the signal is next sent to the speaker.

= Two Times Oversampling =

Oversampling is the process of interpolating data so that it looks like we have more data than we really do. Two times oversampling is accomplished by adding a digital interpolation filter right before the DAC. Now <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math> is convolved with h(t) the desired impulse response. For two times oversampling it would be <math> h(lt/2) = \frac{T}{2} /int </math>

DavidsCD

2004-12-10T10:04:29Z

Andeda:

DavidsCD

2004-12-10T09:45:55Z

Andeda:

CD Players Explained!

insert pic1

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

insert pic2

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC) it is convolved with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math> as shown below.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

In the frequency domain you can see that this relates to multipliing <math> \frac{1}{T} \sum_{n=-\infty}^\infty X(f - \frac{n}{T}) \cdot </math> by P(f) and results in a quite distored X(f). It has to many high frequency components and would require a really good brick wall filter to get rid of them. From there the signal is sent to a Low Pass Filter where the stair stepped shaped function is smoothed so that it sounds better when the signal is next sent to the speaker.

[[Image:barnsasample.jpg|Sampling a signal]]

DavidsCD

2004-12-10T09:40:31Z

Andeda:

CD Players Explained!

insert pic1

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

insert pic2

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC) it is convolved with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math> as shown below.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

In the frequency domain you can see that this relates to multipliing <math> \frac{1}{T} \sum_{n=-\infty}^\infty X(f - \frac{n}{T}) \cdot </math> by P(f) and results in a quite distored X(f). From thier the signal is sent to a Low Pass Filter where the stair stepped shaped function is smoothed so that it sounds better when the signal is next sent to the speaker.

[[Image:barnsasample.jpg|Sampling a signal]]

DavidsCD

2004-12-10T09:34:49Z

Andeda:

DavidsCD

2004-12-10T09:32:36Z

Andeda:

CD Players Explained!

insert pic1

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

insert pic2

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC) it is convolved with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math> as shown below.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

In the frequency domain you can see that this relates to multipliing <math>\tilde \frac{1}{T} \sum_{n=-\infty}^\infty X(f - \frac{n}{T}) \cdot </math> by P(f) and results in a quite distored X(f).

</center>
where
<center>
<math>P(f) = \int_{-\frac{T}{2}}^{\frac{T}{2}} e^{j2\pi ft} \, dt = T sinc(fT)
</math>
</center>

[[Image:barnsasample.jpg|Sampling a signal]]

<math> f(t) = \sum_{k= -\infty}^ \infty \alpha_k e^ \frac{j 2 \pi k t}{T} </math>.

DavidsCD

2004-12-10T09:26:38Z

Andeda:

CD Players Explained!

insert pic1

As seen above storing voice samles on a cd only involves a couple of steps.
First the data must be passed through a low pass filter incase there are any unwanted high frequencies. In our case we would need a filter to pass anything under 22kHz. If we pass anything higher than this then there will be alaising. Next an analog to digital converter (ADC) samples the data at 44000kHz. It does this by basically picking the closest sampling value to the analog value. Next this data is stored on a CD.

insert pic2

Data is taken from the CD player and is represented mathmatically as <math> \sum_{k= -\infty}^ \infty \ x(kT) \delta (t-KT) </math>. When the data goes through the Digital to Analog Converter (DAC)it is convoluted with p(t) to get <math> \tilde x (t) = \sum_{k= -\infty}^ \infty \ x(kT) p (t-KT) </math>.

[[Image:barnsaDA.jpg|Digital to analog conversion]]

[[Image:barnsasample.jpg|Sampling a signal]]

<math> f(t) = \sum_{k= -\infty}^ \infty \alpha_k e^ \frac{j 2 \pi k t}{T} </math>.

DavidsCD

2004-12-10T09:24:48Z

Andeda:

DavidsCD

2004-12-10T08:47:33Z

Andeda:

DavidsCD

2004-12-10T08:38:10Z

Andeda:

DavidsCD

2004-12-10T08:19:25Z

Andeda:

DavidsCD

2004-12-10T08:09:40Z

Andeda:

User:Andeda

2004-11-18T01:39:46Z

Andeda: /* Welcome to David's Wiki Page! */

[[Image:David.jpg|thumb|David's photo]]

== Welcome to David's Wiki Page! ==

Click on the link below to go to my page describing how a CD Player works.

[[DavidsCD]]

User:Andeda

2004-11-18T01:39:14Z

Andeda:

[[Image:David.jpg|thumb|David's photo]]

== Welcome to David's Wiki Page! ==

Click on the link below to go to my CD Player page.

[[DavidsCD]]

User:Andeda

2004-11-18T01:38:40Z

Andeda:

[[Image:David.jpg|thumb|David's photo]]

== Welcome to David's Wiki Page! ==

Click on the link below to go to my CD Player page.
[[DavidsCD]]

User:Andeda

2004-09-28T02:11:26Z

Andeda: /* Welcome to David's Wiki Page! */

[[Image:David.jpg|thumb|David's photo]]

== Welcome to David's Wiki Page! ==
Here there is currently nothing to speak of.

User:Andeda

2004-09-28T02:07:40Z

Andeda:

[[Image:David.jpg|thumb|David's photo]]

== Welcome to David's Wiki Page! ==

User:Andeda

2004-09-28T02:02:23Z

Andeda:

[[Image:David.jpg]]