6.003 signal processing0 two ways to think about dft we can think about the dft in two different...
TRANSCRIPT
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6.003 Signal Processing
Week 5, Lecture B:Discrete Fourier Transform (II)
6.003 Fall 2020
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Discrete Fourier TransformA new Fourier representation for DT signals:
The DFT has a number of features that make it particular convenientβ’ It is not limited to periodic signals. β’ It is discrete in both domains, making it computationally feasible
Synthesis equation
Analysis equation
π₯ π =
π=0
πβ1
π π ππ2πππ π
π π =1
π
π=0
πβ1
π₯[π] β πβπ2πππ π
The FFT (Fast Fourier Transform) is an algorithm for computing the DFT efficiently.
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π₯0 π
Two Ways to Think About DFTWe can think about the DFT in two different ways:
1. Think about DFT as Fourier series of N samples of the signal, periodically extended.
We can see why DFT of a single sinusoid is not concentrated in a single k component
π π =1
π
π=0
πβ1
π₯[π] β πβπ2πππ π
DFT
π0 πiDFT
π₯ π =
π=0
πβ1
π π ππ2πππ π
π₯π π
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Two Ways to Think About DFTWe can think about the DFT in two different ways:
1. Think about DFT as Fourier series of N samples of the signal, periodically extended.
2. Think about DFT as the scaled Fourier transform of a βwindowedβ version of the original signal.
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DFT: Relation to DTFT
ππ€ Ξ© =
π=ββ
β
π₯π€[π] β πβπΞ©π
DTFT
ππ€ Ξ© =
π=0
πβ1
π₯π€[π] β πβπΞ©π
π€[π] = ΰ΅1 0 β€ π < π0 ππ‘βπππ€ππ π
π ππ π =
1
πππ€(
2ππ
π)
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Effect of Windowing on Fourier RepresentationsExample: complex exponential signal π₯ π = ππΞ©0π
Compute the DTFT:
π₯[π] =1
2πΰΆ±2π
π(Ξ©) β ππΞ©π πΞ© = ππΞ©0πWe need to find π(Ξ©) such that :
π₯ π =1
2πΰΆ±βπ
π
π Ξ© β ππΞ©π πΞ© = ππΞ©0π
π Ξ© =
π=ββ
β
2ππΏ(Ξ© β Ξ©0 + 2ππ)
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Effect of Windowing on Fourier Representations
Apply a rectangular window
π₯ π = ππΞ©0πto the complex exponential signal
What the Fourier Transform ππ€ Ξ© look like?
ππ€ Ξ© =
π=ββ
β
π₯π€[π]πβπΞ©π
π€[π] = ΰ΅1 0 β€ π < π0 ππ‘βπππ€ππ π
such that π₯π€ π = π₯ π β π€ π = ππΞ©0ππ€[π]
=
π=ββ
β
ππΞ©0ππ€[π]πβπΞ©π =
π=ββ
β
π€[π]πβπ(Ξ©βΞ©0)π
= π(Ξ© β Ξ©0)
ππΞ©0ππ€[π]π·ππΉπ
π(Ξ© β Ξ©0)
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Effect of Windowing on Fourier Representations
Let As shown below for N=15
What is the DTFT of π€[π]?
π Ξ© =
π=ββ
β
π€[π]πβπΞ©π
π€[π] = ΰ΅1 0 β€ π < π0 ππ‘βπππ€ππ π
=
π=0
πβ1
πβπΞ©π
π Ξ© =1 β πβπΞ©π
1 β πβπΞ©=πβπΞ©
π2(ππΞ©
π2 β πβπΞ©
π2)
πβπΞ©12(ππΞ©
12 β πβπΞ©
12)
=sin(Ξ©
π2)
sin(Ξ©2)πβπΞ©
πβ12
If Ξ©=0, π Ξ© = 0 = π
If Ξ© β 0:
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From Lecture 04B:
ππ(Ξ©) =sin(Ξ©(π +
12)
sin(Ξ©2)
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Effect of Windowing on Fourier Representations
Let As shown below for N=15
What is the DTFT of π€[π]?
π Ξ© =
π=ββ
β
π€[π]πβπΞ©π
π€[π] = ΰ΅1 0 β€ π < π0 ππ‘βπππ€ππ π
=
π=0
πβ1
πβπΞ©π =
π Ξ© = 0
sin(Ξ©π2)
sin(Ξ©2)πβπΞ©
πβ12 , Ξ© β 0
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Effect of Windowing on Fourier Representations
The effect of windowing can be seen from this example of complex exponential signal:
The frequency content of X(β¦) is at discrete frequencies β¦ = β¦o + 2Οm
The frequency content of Xw(β¦) is most dense at these same frequencies, but is spread out over almost all other frequencies as well.
Effect of windowing: spectrum smear
π₯π€ π = ππΞ©0ππ€[π]
π·ππΉπ ππ€ Ξ© = π(Ξ© β Ξ©0)
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DFT: Relation to DTFT
π π =1
π
π=0
πβ1
π₯[π] β πβπ2πππ
π
DFT
ππ€ Ξ© =
π=ββ
β
π₯π€[π] β πβπΞ©π
DTFT
ππ€ Ξ© =
π=0
πβ1
π₯π€[π] β πβπΞ©π
π€[π] = ΰ΅1 0 β€ π < π0 ππ‘βπππ€ππ π
π ππ π =
1
πππ€(
2ππ
π)
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Effect of Windowing on Fourier RepresentationsConsidering the example of π₯ π = ππΞ©0π and π₯π€ π = π
πΞ©0ππ€ π with Ξ©0 =2π
15:
One sample is taken at the peak, and the others fall on zeros.
Because the signal is periodic within the analysis window N.
Ξ© =2ππ
π
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Effect of Windowing on Fourier RepresentationsConsidering the example of π₯ π = ππΞ©0π and π₯π€ π = π
πΞ©0ππ€ π with Ξ©0 =4π
15:
One sample is taken at the peak, and the others fall on zeros.
Because the signal is periodic within the analysis window N.
Ξ© =2ππ
π
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Effect of Windowing on Fourier RepresentationsConsidering the example of π₯ π = ππΞ©0π and π₯π€ π = π
πΞ©0ππ€ π with Ξ©0 =3π
15:
Ξ© =2ππ
π
Now none of the samples fall on zeros.
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Effect of Windowing on Fourier RepresentationsConsidering the example of π₯ π = ππΞ©0π and π₯π€ π = π
πΞ©0ππ€ π with Ξ©0 =2.4π
15:
Ξ© =2ππ
π
Generally, the relation between the samples is complicated.
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Spectral Blurring & Frequency ResolutionLonger windows provide finer frequency resolution.
The width of the central lobe is inversely related to window length.
π Ξ© =sin(Ξ©
π2)
sin(Ξ©2)
πβπΞ©πβ12
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Spectral Blurring & Time/Frequency Tradeoff
β fundamental tradeoff between resolution in frequency and time.
However, longer windows provide less temporal resolution.
π€[π] = ΰ΅1 0 β€ π < π0 ππ‘βπππ€ππ π
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Other types of Windows
π€2 π =ΰ΅π 2 β ΰ΅
π2 β π
ΰ΅π 2
π€1[π]
triangular window
Rectangular window
Hann (or βHanningβ) window
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Effects of Different Types of WindowsThe DFT coefficients of π₯2 π = cos
3π
64π analyzed with N = 64:
Triangular window
Hann (or βHanningβ) window:
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SummaryThe Discrete Fourier Transform (DFT)
The two different perspectives looking at DFT and insights gained
Synthesis equation
Analysis equation
π₯ π =
π=0
πβ1
π π ππ2πππ π
π π =1
π
π=0
πβ1
π₯[π] β πβπ2πππ π
A central issue in using the DFT is understanding the tradeoff between time and frequency
β’ Long analysis windows N provide high resolution in frequency but poor resolution in time.
β’ Short analysis windows N provide high resolution in time but poor resolution in frequency.
The effect of windowing and different types of windows