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Unformatted text preview: shifting, and reversal can all be combined.
• Operation can be performed in any order, but care is required.
• This will cause confusion.
• Example: x(2(t − 1))
Scale ﬁrst, then shift
Compress by 2, shift by 1
2 2
1
2 0 1 x(2t ) x(t )
1 2 1 1 0 2 t x(2(t − 1))
1 2 2 t 1 2
1
0 1 2 t EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 9 Example x(2(t − 1)), continued
Shift ﬁrst, then scale
Shift by 1, compress by 2 Incorrect
2 2
1
2 0 1 x(t − 1) x(t )
1 2 1 0 2 t 2 1 x(2(t − 1))
1 2 2 t 1 1
0 1 Shift ﬁrst, then scale
Rewrite x(2(t − 1)) = x(2t − 2)
Shift by 2, scale by 2
1
2 1 0 1 x(t − 2) x(t )
2 t 2 t Correct
2 2 2 1 0 2 x(2t − 2) 1
1 2 3 t 2 1 1
0 1 2 t Where is 2(t − 1) equal to zero?
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 10 Try these yourselves ....
1 x(t )
4 2 0 x(2(t + 2)) 4 2 2 4 t 4 2 0 x(−t + 1) 1 0 1 x(−t /2) 2 4 t 4 2 2 4 t 2 4 t 1 0 EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 11 Even and Odd Symmetry • An even signal is symmetric about the origin
x(t) = x(−t)
• An odd signal is antisymmetric about the origin
x(t) = −x(−t)
2 x(t ) = x(−t )
2 1 1
0 2 Even
1 x(t ) = −x(−t )
2 t EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 2 1 Odd 1
0 1 2 t 12 • Any signal can be decomposed into even and odd components
1
[x(t) + x(−t)]
2
1
[x(t) − x(−t)] .
2 xe(t) =
xo(t) =
Check that xe(t) = xe(−t), xo(t) = −xo(−t),
and that xe(t) + xo(t) = x(t). EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 13 • Example x(t )
1 2 2 1 1 x(−t ) 0 1 t 1 t 0 1 t 1 2
1
1 2
1 0 1 t 1
xe(t ) = [x(t ) + x(−t )]
2 1 0 1
xo(t ) = [x(t ) − x(−t )]
2 • Same type of decomposition applies for discretetime signals.
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 14 The decomposition into even and odd components depends on the location
of the origin. Shifting the signal changes the decomposition.
Plot the even and odd components of the previous example, after shifting
x(t) by 1/2 to the right.
2 1 2 1 x(t ) 1 x(−t ) 0 1 0 t 1 1 2 1 2 1 t 1 0 t 1 1
xe(t ) = [x(t ) + x(−t )]
2 1 0 1 t 1
xo(t ) = [x(t ) − x(−t )]
2 EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 15 Discrete Amplitude Signals • Discrete amplitude signals take on only a countable set of values.
• Example: Quantized signal (binary, ﬁxed point, ﬂoating point).
• A digital signal is a quantized discretetime signal.
• Requires treatment as random process, not part of this course.
x(t ) 4
2
2 1 0
2 x[n] 4
2 1 2 t EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 2 1 0 1 2 t 2 16 Periodic Signals
• Very important in this class.
• Continuous time signal is periodic if and only if there exists a T0 > 0
such that
x(t + T0) = x(t)
for all t
T0 is a period of x(t) in time.
• A discretetime signal is periodic if and only if there exists an intege...
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This note was uploaded on 09/28/2013 for the course EE 108b taught by Professor Mukamel during the Fall '13 term at Singapore Stanford Partnership.
 Fall '13
 Mukamel
 Signal Processing

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