Fourier transform math - Revision history

imported>Psych204B: /* Phase and amplitude */

2012-03-16T04:11:38Z

Phase and amplitude

← Older revision		Revision as of 04:11, 16 March 2012
Line 57:		Line 57:
	If a complex number is just a point in space, we can also write it in polar coordinates: a radius r, extending from the origin to a+bi, and an angle theta, the angle between the real axis and the radius r. The radius r is the amplitude of the sine wave and the angle theta is its phase.		If a complex number is just a point in space, we can also write it in polar coordinates: a radius r, extending from the origin to a+bi, and an angle theta, the angle between the real axis and the radius r. The radius r is the amplitude of the sine wave and the angle theta is its phase.

	[[File: ph.jpg \| ~~200~~ px]]		[[File: ph.jpg \| 300 px]]

imported>Psych204B: /* Phase and amplitude */

2012-03-16T04:11:11Z

Phase and amplitude

← Older revision		Revision as of 04:11, 16 March 2012
Line 52:		Line 52:

	===Phase and amplitude===		===Phase and amplitude===
	[[~~Image~~: complex.jpg \| right \| 150 px]]		[[File: complex.jpg \| right \| 150 px]]
	The Fourier coefficient is kind of a mysterious number. It contains both phase and amplitude information, but how do we get it? First, we know the Fourier coefficient is a complex number a + bi. Imaginary numbers are kind of confusing, but they're really just another "axis," like the x-axis or y-axis. So a complex number just represents a point in the complex plane, where one axis is the real component while the other is the imaginary component.		The Fourier coefficient is kind of a mysterious number. It contains both phase and amplitude information, but how do we get it? First, we know the Fourier coefficient is a complex number a + bi. Imaginary numbers are kind of confusing, but they're really just another "axis," like the x-axis or y-axis. So a complex number just represents a point in the complex plane, where one axis is the real component while the other is the imaginary component.

			If a complex number is just a point in space, we can also write it in polar coordinates: a radius r, extending from the origin to a+bi, and an angle theta, the angle between the real axis and the radius r. The radius r is the amplitude of the sine wave and the angle theta is its phase.

			[[File: ph.jpg \| 200 px]]

imported>Psych204B: /* Phase and amplitude */

2012-03-16T04:06:18Z

Phase and amplitude

← Older revision		Revision as of 04:06, 16 March 2012
Line 52:		Line 52:

	===Phase and amplitude===		===Phase and amplitude===
	[[Image: complex.jpg \| right]]		[[Image: complex.jpg \| right \| 150 px]]
	The Fourier coefficient is kind of a mysterious number. It contains both phase and amplitude information, but how do we get it? First, we know the Fourier coefficient is a complex number a + bi. Imaginary numbers are kind of confusing, but they're really just another "axis," like the x-axis or y-axis. So a complex number just represents a point in the complex plane, where one axis is the real component while the other is the imaginary component.		The Fourier coefficient is kind of a mysterious number. It contains both phase and amplitude information, but how do we get it? First, we know the Fourier coefficient is a complex number a + bi. Imaginary numbers are kind of confusing, but they're really just another "axis," like the x-axis or y-axis. So a complex number just represents a point in the complex plane, where one axis is the real component while the other is the imaginary component.

imported>Psych204B: /* Phase and amplitude */

2012-03-16T04:05:53Z

Phase and amplitude

← Older revision		Revision as of 04:05, 16 March 2012
Line 51:		Line 51:
	F(w) is just the set of all the Fourier coefficients for all the sinusoids at different frequencies. How many different frequencies? The Fourier series says that any repeating signal can be written as an '''infinite''' number of sinusoids. However, in fMRI, we are discretely sampling our data (every TR, typically ~2 sec), so there is an upper limit to the number of frequencies we can use: #times sampled/2. This is the [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist frequency], which sets the maximum resolution for a discretely sampled signal. So we don't have an infinite number of Fourier coefficients. We have #times/smpled + 1 coefficients.		F(w) is just the set of all the Fourier coefficients for all the sinusoids at different frequencies. How many different frequencies? The Fourier series says that any repeating signal can be written as an '''infinite''' number of sinusoids. However, in fMRI, we are discretely sampling our data (every TR, typically ~2 sec), so there is an upper limit to the number of frequencies we can use: #times sampled/2. This is the [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist frequency], which sets the maximum resolution for a discretely sampled signal. So we don't have an infinite number of Fourier coefficients. We have #times/smpled + 1 coefficients.

	==Phase and amplitude==		===Phase and amplitude===
			[[Image: complex.jpg \| right]]
			The Fourier coefficient is kind of a mysterious number. It contains both phase and amplitude information, but how do we get it? First, we know the Fourier coefficient is a complex number a + bi. Imaginary numbers are kind of confusing, but they're really just another "axis," like the x-axis or y-axis. So a complex number just represents a point in the complex plane, where one axis is the real component while the other is the imaginary component.

imported>Psych204B: /* Discrete Fourier Transform */

2012-03-16T03:57:00Z

Discrete Fourier Transform

← Older revision		Revision as of 03:57, 16 March 2012
Line 50:		Line 50:

	F(w) is just the set of all the Fourier coefficients for all the sinusoids at different frequencies. How many different frequencies? The Fourier series says that any repeating signal can be written as an '''infinite''' number of sinusoids. However, in fMRI, we are discretely sampling our data (every TR, typically ~2 sec), so there is an upper limit to the number of frequencies we can use: #times sampled/2. This is the [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist frequency], which sets the maximum resolution for a discretely sampled signal. So we don't have an infinite number of Fourier coefficients. We have #times/smpled + 1 coefficients.		F(w) is just the set of all the Fourier coefficients for all the sinusoids at different frequencies. How many different frequencies? The Fourier series says that any repeating signal can be written as an '''infinite''' number of sinusoids. However, in fMRI, we are discretely sampling our data (every TR, typically ~2 sec), so there is an upper limit to the number of frequencies we can use: #times sampled/2. This is the [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist frequency], which sets the maximum resolution for a discretely sampled signal. So we don't have an infinite number of Fourier coefficients. We have #times/smpled + 1 coefficients.

			==Phase and amplitude==

imported>Psych204B: /* Finding the Fourier coefficient */

2012-03-16T03:45:15Z

Finding the Fourier coefficient

← Older revision		Revision as of 03:45, 16 March 2012
Line 36:		Line 36:
	So now we've got a bunch of sine and cosine waves (written using Euler's formula). How do we find the Fourier coefficient which tells us how much to "weight" each wave? Let f(t) be the repeating waveform, then for any sine/cosine wave e^i2pi*nft:		So now we've got a bunch of sine and cosine waves (written using Euler's formula). How do we find the Fourier coefficient which tells us how much to "weight" each wave? Let f(t) be the repeating waveform, then for any sine/cosine wave e^i2pi*nft:

	[[File: coeff.jpg \| ~~150~~]]		[[File: coeff.jpg \| 200 px]]

	This equation also looks really intimidating! But remember that the e^whatsit is just a sinusoid at a certain frequency, f(t) is the repeating wave form, and c is just a complex number of the form a + bi. The integral from negative infinity to positive infinity is just a mathematical way of saying, "Let's find the correlation between our repeating waveform and some sinusoid." A large correlation indicates that a sinusoid really characterizes the repeating waveform, which is what I previously said the coefficient says.		This equation also looks really intimidating! But remember that the e^whatsit is just a sinusoid at a certain frequency, f(t) is the repeating wave form, and c is just a complex number of the form a + bi. The integral from negative infinity to positive infinity is just a mathematical way of saying, "Let's find the correlation between our repeating waveform and some sinusoid." A large correlation indicates that a sinusoid really characterizes the repeating waveform, which is what I previously said the coefficient says.

imported>Psych204B: /* Finding the Fourier coefficient */

2012-03-16T03:45:04Z

Finding the Fourier coefficient

← Older revision		Revision as of 03:45, 16 March 2012
Line 36:		Line 36:
	So now we've got a bunch of sine and cosine waves (written using Euler's formula). How do we find the Fourier coefficient which tells us how much to "weight" each wave? Let f(t) be the repeating waveform, then for any sine/cosine wave e^i2pi*nft:		So now we've got a bunch of sine and cosine waves (written using Euler's formula). How do we find the Fourier coefficient which tells us how much to "weight" each wave? Let f(t) be the repeating waveform, then for any sine/cosine wave e^i2pi*nft:

	[[File: coeff.jpg \| ~~200~~]]		[[File: coeff.jpg \| 150]]

	This equation also looks really intimidating! But remember that the e^whatsit is just a sinusoid at a certain frequency, f(t) is the repeating wave form, and c is just a complex number of the form a + bi. The integral from negative infinity to positive infinity is just a mathematical way of saying, "Let's find the correlation between our repeating waveform and some sinusoid." A large correlation indicates that a sinusoid really characterizes the repeating waveform, which is what I previously said the coefficient says.		This equation also looks really intimidating! But remember that the e^whatsit is just a sinusoid at a certain frequency, f(t) is the repeating wave form, and c is just a complex number of the form a + bi. The integral from negative infinity to positive infinity is just a mathematical way of saying, "Let's find the correlation between our repeating waveform and some sinusoid." A large correlation indicates that a sinusoid really characterizes the repeating waveform, which is what I previously said the coefficient says.

imported>Psych204B: /* Discrete Fourier Transform */

2012-03-16T03:44:33Z

Discrete Fourier Transform

← Older revision		Revision as of 03:44, 16 March 2012
Line 47:		Line 47:
	The frequency domain is a way of representing the original signal in terms of its frequency components. Instead of time on the x-axis, we have frequency. Instead of signal magnitude on the y-axis, we have a measure of the power at the frequency, which is just the amplitude of the sinusoid at that frequency. how do we get these amplitudes? Remember how we found the Fourier coefficients in the previous section? Finding that coefficient at a particular frequency IS taking the Fourier transform at that frequency. Taking the Fourier transform of a signal is just finding the Fourier coefficient for all the sine waves at all the possible frequencies:		The frequency domain is a way of representing the original signal in terms of its frequency components. Instead of time on the x-axis, we have frequency. Instead of signal magnitude on the y-axis, we have a measure of the power at the frequency, which is just the amplitude of the sinusoid at that frequency. how do we get these amplitudes? Remember how we found the Fourier coefficients in the previous section? Finding that coefficient at a particular frequency IS taking the Fourier transform at that frequency. Taking the Fourier transform of a signal is just finding the Fourier coefficient for all the sine waves at all the possible frequencies:

	[[transform.jpg \| 200 px]]		[[File: transform.jpg \| 200 px]]

	F(w) is just the set of all the Fourier coefficients for all the sinusoids at different frequencies. How many different frequencies? The Fourier series says that any repeating signal can be written as an '''infinite''' number of sinusoids. However, in fMRI, we are discretely sampling our data (every TR, typically ~2 sec), so there is an upper limit to the number of frequencies we can use: #times sampled/2. This is the [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist frequency], which sets the maximum resolution for a discretely sampled signal. So we don't have an infinite number of Fourier coefficients. We have #times/smpled + 1 coefficients.		F(w) is just the set of all the Fourier coefficients for all the sinusoids at different frequencies. How many different frequencies? The Fourier series says that any repeating signal can be written as an '''infinite''' number of sinusoids. However, in fMRI, we are discretely sampling our data (every TR, typically ~2 sec), so there is an upper limit to the number of frequencies we can use: #times sampled/2. This is the [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist frequency], which sets the maximum resolution for a discretely sampled signal. So we don't have an infinite number of Fourier coefficients. We have #times/smpled + 1 coefficients.

imported>Psych204B: /* Discrete Fourier Transform */

2012-03-16T03:44:11Z

Discrete Fourier Transform

← Older revision		Revision as of 03:44, 16 March 2012
Line 43:		Line 43:

	==Discrete Fourier Transform==		==Discrete Fourier Transform==
			So then what is a Fourier Transform? The Fourier Transform takes a waveform in the time domain and transforms it to the frequency domain. The time domain is typically how we think of our signal (repeating waveform, etc): at different points in time, the signal has a different magnitude. If we plotted the signal in the time domain, the x-axis would be time and the y axis would be some value of the signal.

			The frequency domain is a way of representing the original signal in terms of its frequency components. Instead of time on the x-axis, we have frequency. Instead of signal magnitude on the y-axis, we have a measure of the power at the frequency, which is just the amplitude of the sinusoid at that frequency. how do we get these amplitudes? Remember how we found the Fourier coefficients in the previous section? Finding that coefficient at a particular frequency IS taking the Fourier transform at that frequency. Taking the Fourier transform of a signal is just finding the Fourier coefficient for all the sine waves at all the possible frequencies:

			[[transform.jpg \| 200 px]]

			F(w) is just the set of all the Fourier coefficients for all the sinusoids at different frequencies. How many different frequencies? The Fourier series says that any repeating signal can be written as an '''infinite''' number of sinusoids. However, in fMRI, we are discretely sampling our data (every TR, typically ~2 sec), so there is an upper limit to the number of frequencies we can use: #times sampled/2. This is the [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist frequency], which sets the maximum resolution for a discretely sampled signal. So we don't have an infinite number of Fourier coefficients. We have #times/smpled + 1 coefficients.

imported>Psych204B: /* Finding the Fourier coefficient */

2012-03-16T03:19:22Z

Finding the Fourier coefficient

← Older revision		Revision as of 03:19, 16 March 2012
Line 41:		Line 41:

	How does finding an integral of a product give you a measure of correlation? From high school, you know that you integrate to find the area under a curve. A really simple of way of thinking about this equation then is that if a sinusoid and the waveform are correlated, then the product of the areas under their curves will be positive. If that didn't make sense, watch [http://www.youtube.com/watch?v=ObklYbQaX24 this guy's] video. He explains it much better.		How does finding an integral of a product give you a measure of correlation? From high school, you know that you integrate to find the area under a curve. A really simple of way of thinking about this equation then is that if a sinusoid and the waveform are correlated, then the product of the areas under their curves will be positive. If that didn't make sense, watch [http://www.youtube.com/watch?v=ObklYbQaX24 this guy's] video. He explains it much better.

			==Discrete Fourier Transform==