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![Page 1: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/1.jpg)
DTFT continue (c.f. Shenoi, 2006)
We have introduced DTFT and showed some of its properties. We will investigate them in more detail by showing the associated derivations later.
We have also given a motivation of DFT which is both discrete in time and frequency domains. We will also introduce DFT in more detail below.
![Page 2: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/2.jpg)
DC responseWhen w=0, the complex exponential e−jw becomes a constant
signal, and the frequency response X(ejw) is often called the DC response when w=0.
–The term DC stands for direct current, which is a constant current.
![Page 3: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/3.jpg)
We will represent the spectrum of DTFT either by H(ejwT) or more often by H(ejw) for convenience.
When represented as H(ejw), it has the frequency range [-π,π]. In this case, the frequency variable is to be understood as the normalized frequency. The range [0, π] corresponds to [0, ws/2] (where wsT=2π), and the normalized frequency πcorresponds to the Nyquist frequency (and 2π corresponds to the sampling frequency).
![Page 4: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/4.jpg)
DC responseWhen w=0, the complex exponential e−jw becomes a constant
signal, and the frequency response X(ejw) is often called the DC response when w=0.
–The term DC stands for direct current, which is a constant current.
DTFT Properties Revisited
Time shifting
![Page 5: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/5.jpg)
Frequency shifting
Time reversal
![Page 6: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/6.jpg)
DTFT of δ(n)
DTFT of δ(n+k)+ δ(n-k)According to the time-shifting property,
Hence
( ) jwnn
nen −
−∞=∑δ
( ) ( ) jwkjwk eknekn −−+ is of DTFT , is of DTFT δδ
( ) ( ) ( )wkeeknkn jwkjwk cos2 is of DTFT =+−++ −δδ
10 == jwe
![Page 7: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/7.jpg)
DTFT of x(n) = 1 (for all n)x(n) can be represented as
We prove that its DTFT is
![Page 8: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/8.jpg)
Hence
( ) dwekw jwn
k∫ ∑−
∞
−∞=⎥⎦
⎤⎢⎣
⎡−
π
ππδ 2
( ) ( ) ( ) ( )dwwdwwdwwdww ∫∫∫∫∞
−
−
∞−−++==
π
π
π
ππ
πδδδδ
( ) dwkwk
∫ ∑−
∞
−∞=⎥⎦
⎤⎢⎣
⎡−=
π
ππδ 2
( ) ndww allfor 1== ∫∞
∞−δ
![Page 9: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/9.jpg)
From another point of viewAccording to the sampling property: the DTFT of a continuous
signal xa(t) sampled with period T is obtained by a periodic duplication of the continuous Fourier transform Xa(jw) with a period 2π/T = ws and scaled by T.Since the continuous F.T. of x(t)=1 (for all t) is δ(t), the DTFT of x(n)=1 shall be a impulse train (or impulse comb), and it turns out to be
DTFT of anu(n) (|a|<1)let
then
This infinite sequence converges to when |a|<1.
![Page 10: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/10.jpg)
DTFT of Unit Step Sequence
![Page 11: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/11.jpg)
Adding these two results, we have the final result
![Page 12: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/12.jpg)
Differentiation Property
![Page 13: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/13.jpg)
DTFT of a rectangular pulse
![Page 14: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/14.jpg)
and get
![Page 15: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/15.jpg)
ConvolutionConvolution of two discrete-time signalsLet x[n] and h[n] be two signals, the convolution of x and h is
can be written in short by y[n] = x[n] ∗ h[n].
Convolution of two continuous-time signals
can be written in short by y(t) = x(t) ∗ h(t)
[ ] [ ] [ ]∑∞
−∞=
−=k
knhkxny
( ) ( ) ( )∫∞
∞−
−= τττ dthxty
![Page 16: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/16.jpg)
Continuous convolution: optics example
From Kuhn 2005
![Page 17: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/17.jpg)
Continuous convolution: electronics example
From Kuhn 2005
![Page 18: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/18.jpg)
Properties of convolutionCommunitive
x[n] ∗ h[n] = h[n] ∗ x[n]this means that y[n] can also be represented as
Associativex[n] ∗ (h1[n] ∗ h2[n]) = (x[n] ∗ h1[n]) ∗ h2[n].
Linearx[n] ∗ (ah1[n] + bh2[n]) = ax[n] ∗ h1[n] + bx[n] ∗ h2[n].
Sequence shifting is equivalent to convolute with a shifted impulse
x[n-d]= x[n] ∗ δ[n-d]
[ ] [ ] [ ]∑∞
−∞=
−=k
khknxny
![Page 19: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/19.jpg)
An illustrative example
x[n]∗δ[n] x[n]∗h[n]
![Page 20: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/20.jpg)
![Page 21: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/21.jpg)
![Page 22: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/22.jpg)
Convolution can be realized by–Reflecting h[k] about the origin to obtain h[-k].–Shifting the origin of the reflected sequences to k=n.–Computing the weighted moving average of x[k] by using the weights given by h[n-k].
![Page 23: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/23.jpg)
![Page 24: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/24.jpg)
![Page 25: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/25.jpg)
Convolution can be explained as “arithmetic product.”•Eg.,
–x[n] = 0, 0, 5, 2, 3, 0, 0…–h[n] = 0, 0, 1, 4, 3, 0, 0…–x[n] ∗ h[n]:
0 9, 22,26,18, 5, 0, 0,
0 9, 6, 15, 0, 0, 0, 0, 0 0, 12, 8, 20, 0, 0, 0, 0 0, 0, 3, 2, 5 0, 0,
... 0, 0, 3, 4, 1, 0, 0, *)... 0, 0, 3, 2, 5 0, 0
,
,,
![Page 26: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/26.jpg)
Convolution vs. Fourier TransformMultiplication Property: For continuous F.T. and DTFT, if we perform multiplication in time domain, then it is equivalent to performing convolution in the frequency domain, and vice versa.
DTFT convolution theorem:Let x[n] ↔ X(ejw) and h[n] ↔ H(ejw).If y[n] = x[n] ∗ h[n], then
Y(ejw) = X(ejw)H(ejw)modulation/windowing theorem (or multiplication property)
Let x[n] ↔ X(ejw) and w[n] ↔ W(ejw).If y[n] = x[n]w[n], then
( ) ( ) ( )( )∫−
−=π
π
θθ θπ
deWeXeY wjjjw
21 a periodic
convolution
![Page 27: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/27.jpg)
For continuous F. T.
In summary,
![Page 28: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/28.jpg)
Sampling theorem revisited (oppenheim et al. 1999)
An ideal continuous-to-discrete-time (C/D) converter
Let s(t) be a continuous signal, which is a periodic impulse train:
We modulate s(t) with xc(t), obtaining
( ) ( )∑∞
∞−
−= nTtts δ
( ) ( ) ( ) ( ) ( )∑∞
∞−
−== nTttxtstxtx ccs δ
![Page 29: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/29.jpg)
Examples of xs(t) for two sampling rates
![Page 30: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/30.jpg)
Through the ‘sifting property’ of the impulse function, xs(t) can be expressed as
Let us now consider the continuous Fourier transform of xs(t). Since xs(t) is a product of xc(t) and s(t), its continuous Fourier transform is the convolution of Xc(jΩ) and S(jΩ).
( ) ( ) ( )∑∞
∞−
−= nTtnTxtx cs δ
Sampling with a periodic impulse followed by conversion to a discrete-time sequence
![Page 31: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/31.jpg)
Note that the continuous Fourier transform of a periodic impulse train is a periodic impulse train.
where Ωs =2π/T is the sampling frequency in radians/s.
Since
If follows that
Again, we see that the copies of Xc(jΩ) are shifted by integer multiples of the sampling frequency, and then superimposed to product the periodic Fourier transform of the impulse train of samples.
( ) ( ) ( )Ω∗Ω=Ω jSjXjX cs
( ) ( )( )∑∞
−∞=
Ω−Ω=Ωk
scs kjXT
jX 1
( ) ( )∑∞
−∞=
Ω−Ω=Ωk
skT
jS δ1
![Page 32: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/32.jpg)
![Page 33: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/33.jpg)
![Page 34: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/34.jpg)
![Page 35: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/35.jpg)
Sampling of Bandpass Signals (c.f. Shenoi, 2006)
![Page 36: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/36.jpg)
Correlation
Given a pair of sequences x[n] and y[n], their cross correlation sequence is rxy[l] is defined as
for all integer l. The cross correlation sequence can sometimes help to measure similarities between two signals.
It’s very similar to convolution, unless the indices changes from l − n to n − l.
Autocorrelation:
[ ] [ ] [ ] [ ] [ ]lynxlnynxlrn
xy −∗=−= ∑∞
−∞=
[ ] [ ] [ ]∑∞
−∞=
−=n
xx lnxnxlr
![Page 37: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/37.jpg)
Consider the following non-negative expression:
That is,
Thus, the matrix is positive semidefinite.
Its determinate is nonnegative.
Properties
[ ] [ ]( ) [ ] [ ] [ ] [ ]
[ ] [ ] [ ] 0020
2
2
2222
≥++=
−+−+=−+ ∑∑∑∑∞
−∞=
∞
−∞=
∞
−∞=
∞
−∞=
yyxyxx
nnnn
rlarra
lnylnynxanxalnynax
[ ] [ ] [ ][ ] [ ] a
arlr
lrra
yyxy
xyxx allfor 010
01 ≥⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
[ ] [ ][ ] [ ]⎥⎦
⎤⎢⎣
⎡0
0
yyxy
xyxx
rlrlrr
![Page 38: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/38.jpg)
The determinant is rxx[0]ryy[0] − rxy2[l] ≥ 0.
Propertiesrxx[0]ryy[0] ≥ rxy
2[l]rxx
2[0] ≥ rxy2[l]
Normalized cross correlation and autocorrelation:
The properties imply that |ρxx[0]|≤1 and |ρyy[0]|≤1.The DTFT of the autocorrelation signal rxx[l] is the squared magnitude of the DTFT of x[n], i.e., |X(ejw)|2.
Correlation is useful in random signal processing
[ ] [ ][ ] [ ] [ ]
[ ] [ ]00
0 yyxx
xyxy
xx
xxxx
rr
lrl
rlr
l == ρρ
![Page 39: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/39.jpg)
DFT and DTFT – A closer lookWe discuss the DTFT-IDTFT pair (‘I’ means “inverse) for a discrete-time function given by
and
The pair and their properties and applications are elegant, but from practical point of view, we see some limitations; eg. the input signal is usually aperiodic and may be finite in length.
![Page 40: DTFT continue (c.f. Shenoi, 2006) - 國立臺灣大學cmlab.csie.ntu.edu.tw/DSPCourse/slide/Lecture3.pdf · 2006-11-06 · Convolution Convolution of two discrete-time signals Let](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b778b847f8b9a805c8d3e20/html5/thumbnails/40.jpg)
Example of a finite-length x(n) and its DTFT X(ejw).
A finite-length signal
Its magnitude spectrum
Its phase spectrum
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The function X(ejw) is continuous in w, and the integration is not suitable for computation by a digital computer.
We can discretize the frequency variable and find discrete values for X(ejwk), where wk are equally sampled whthin [-π, π].
Discrete-time Fourier Series (DFS)Let x(n) (n∈Z) be a finite-length sequence, with the length being N; i.e., x(n) = 0 for n < 0 and n > N.Consider a periodic expansion of x(n):
xp(n) is periodic, so it can be represented as a Fourier series:( ) ( ) integerany is ,1,...,1,0 , KnnnxKNnxp −==+
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To find the coefficients Xp(k) (with respect to a discrete periodic signal), we use the following summation, instead of integration:
First, multiply both sides by e-jmw0k, and sum over n from n=0 to n=N-1:
By interchanging the order of summation, we get
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Noting that
pf. When n=m, the summation reduces to
N
When n≠m, by applying the geometric-sequence formula,
we have
( ) ( ) ( )( ) ( )
( )
( ) ( ) ( )
( ) 01
1
/2
)(/2)(/22
/2
)(/2)(/21
'
'/21
0
/2
=−−
=
−−
==
−−+
−−−−
−=
−
=
− ∑∑
kNj
mkNjmkNjkj
kNj
mkNjmNkNjmN
mn
knNjN
n
mnkNj
eee
eeeee
π
πππ
π
ππππ
1 ,1
1
≠−−
=−+
−=∑ r
rrrr
MNN
Mn
n
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Since there is only one nonzero term,
= Xp(k)N
The final result is
The following pairs then form the DFS
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Relation between DTFT and DFS for finite-length sequences
We note that
In other words, when the DTFT of the finite length sequence x(n) is evaluated at the discrete frequency wk = (2π/N)k, (which is the kth sample when the frequency range [0, 2π] is divided into N equally spaced points) and dividing by N, we get the Fourier series coefficients Xp(k).
DTFT spectrumDFS coeficient
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A finite-length signal
Its magnitude spectrum (sampled)
Its phase spectrum (sampled)
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To simplify the notation, let us denote
The DFS-IDFS (‘I’ means “inverse”) can be rewritten as (W=WN)
Discrete Fourier Transform (DFT)Consider both the signal and the spectrum only within one
period (N-point signals both in time and frequency domains)
( )NjN eW /2π−=
DFT
IDFT(inverseDFT)
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Relation between DFT and DTFT: The frequency coefficients of DFT is the N-point uniform samples of DTFT with [0, 2π].
The two equations DFT and IDFT give us a numerical algorithm to obtain the frequency response at least at the N discrete frequencies. By choosing a large value N, we get a fairly good idea of the frequency response for x(n), which is a function of the continuous variable w.
Question: Can we reconstruct the DTFT spectrum (continuous in w) from the DFT?
→ Since the N-length signal can be exactly recovered from both the DFT coefficients and the DTFT spectrum, we expect that the DTFT spectrum (that is within [0, 2π]) can be exactly reconstructed by the DCT coefficients.
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Reconstruct DTFT from DFT(when the sequence is finite-length)
By substituting the inverse DFT into the x(n), we have
a geometric sequence
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By applying the geometric-sequence formula
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So
The reconstruction formula