6.003 Signal Processing0 Two Ways to Think About DFT We can think about the DFT in two different...
Transcript of 6.003 Signal Processing0 Two Ways to Think About DFT We can think about the DFT in two different...
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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