Embed Size (px)
Citation preview
DSP First 2/e
Lecture 5A:Operations on the Spectrum
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 2
License Info for DSPFirst Slides
§ This work released under a Creative Commons Licensewith the following terms:
§ Attribution§ The licensor permits others to copy, distribute, display, and perform
the work. In return, licensees must give the original authors credit.
§ Non-Commercial§ The licensor permits others to copy, distribute, display, and perform
the work. In return, licensees may not use the work for commercial purposes—unless they get the licensor's permission.
§ Share Alike§ The licensor permits others to distribute derivative works only under
a license identical to the one that governs the licensor's work.§ Full Text of the License§ This (hidden) page should be kept with the presentation
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 3
READING ASSIGNMENTS
§ This Lecture:§ Chapter 3, Section 3-3 (DSP-First 2/e)
§ Other Reading:§ Appendix A: Complex Numbers
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 4
LECTURE OBJECTIVES
§ Operations on a time-domain signal x(t) have a SIMPLE form in the frequency-domain
§ SPECTRUM Representation has lines at:
§ Represents Sinusoid with DIFFERENT Frequencies
å=
+=N
kkkk tfAtx
1)2cos()( jp
),,( kkk fA j
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 5
Recall FREQUENCY DIAGRAM
§ Used to visualize relationship between frequencies, amplitudes and phases
§ Plot Complex Amplitude vs. Freq
0 100 250–100–250 f (in Hz)
3/7 pje 3/7 pje-2/4 pje- 2/4 pje
10 Complex amplitude
Spectral line
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 6
GRAPHICAL SPECTRUM
AMPLITUDE, PHASE & FREQUENCY are shown
w7-7 0
)6.07cos(2
22)1.07sin(276.076.0
71.05.02171.05.0
21
p
ppp
pppppp
+=+=
+=+---
----
teeee
eeeeeeeettjjtjj
tjjjjtjjjj
p6.0je- p6.0jeFreq. in rad/s
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 7
§ 2M + 1 spectrum components:
§ At the complex amplitude is
§ usually, for real x(t)
General Spectrum
å-=
=M
Mk
tfjk
keatx p2)(
kakff =00 =f
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 8
§ Adding DC, or amplitude scaling§ Adding two (or more) signals§ Time-Shifting
§ Multiply in frequency by complex exponential§ Differentiation of x(t)
§ Multiply in frequency-domain by (jw)§ Frequency Shifting
§ Multiply in time-domain by sinusoid
OPERATIONS on SPECTRUM
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 9
§ Adding DC
§ Scaling
Scaling or Adding a constant
åå-=-=
==M
Mk
tfjk
M
Mk
tfjk
kk eaeatx pp ggg 22 )()(
!!! "!!! #$ca
tjtj
k
tfjk ceeaeactx k
+¹
++=+ å0 is new DC
)0(2)0(20
0
2)( ppp
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 10
Scaling and Adding a constant
!"#new DC
00
2 6226)(2 ++=+ å¹
aeatxk
tfjk
kp
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 11
§ Adding signals with same fundamental
Adding Two Signals (1)
ååå-=-=-=
+=+=+M
Mk
tfjkk
M
Mk
tfjk
M
Mk
tfjk
kkk eaaeaeatxtx ppp 221
22
2121 )()()(
?)(
)(
2
1
tx
tx
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 12
§ Adding signals with same fundamental
Adding Two Signals (2)
)()(
)(
)(
21
2
1
txtx
tx
tx
+
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 13
§ Time Shifting
§ Multiply Spectrum complex amplitudes by a complex exponential
Time Shifting x(t)
å
åå
-=
-=
-
-=
-
=
==-
M
Mk
tfjk
M
Mk
tfj
b
fjk
M
Mk
tfjkd
k
k
k
dkdk
ebty
eeaeatx
p
ptptpt
2
22)(2
)(
)()( !"!#$
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 14
§ Take derivative of the Signal x(t)
§ Multiply complex amplitudes by “jw”=“j2pf”
Differentiating x(t)
å
åå
-=
-=-=
=
==
M
Mk
tfjk
M
Mk
tfj
b
kk
M
Mk
tfjkkdt
d
k
k
k
k
ebty
eafjefjatx
p
pp pp
2
22
)(
)2()2()( !"!#$
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 15
§ Multiply x(t) by Complex Exponential èFrequency Shifting
§ Spectrum components shifted:
Frequency Shifting x(t)
å
å
-=
+
-=
=
=
=
M
Mk
tffjjk
M
Mk
tfjk
tfjj
tfjj
ck
kc
c
eAea
eaeAety
txeAety
)(2
22
2
)(
)(
)()(
pj
ppj
pj
ckk fff +®
5/26/2016 © 2015-2016, JH McClellan & RW Schafer 16
Frequency Shifting x(t)
))9(2sin()(
)(
)(
Hz9by upShift
)9(2
ttx
etx
tx
tj
p
p
