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Unformatted text preview: ber of terms compresses ringing, but doesn’t reduce its
amplitude! EE102A:Signal Processing and Linear Systems I; Win 0809 Pauly 16 Impulse Trains
One of the most important signals we talked about was the impulse train,
δT (t) = ∞
n=−∞ δ (t − nT ) These are selftransforming
δT (t) ⇔ ω0δω0 (ω )
The Fourier transform of an array evenly spaced δ ’s is another array of
evenly spaced δ ’s!
Connects Fourier series and Fourier transform.
Models sampling.
EE102A:Signal Processing and Linear Systems I; Win 0809 Pauly 17 Periodic Sequences
If f1(t) is one cycle of a periodic function with period T , then f (t) is
f (t) = f1(t) ∗ δT (t)
and the Fourier transform is
F (j ω ) = ω0F1(j ω )δω0 (ω )
Multiplying by δω0 (ω ) samples F1(j ω ) at multiples of ω0.
The Fourier transform of the periodic signal is the sampled Fourier transform
of one period EE102A:Signal Processing and Linear Systems I; Win 0809 Pauly 18 Square wave, period 2. Convolution of rect(t) and the impulse train δ2(t)
rect(t )
−3 −3 −2 −2 −1 −1 0 1 ∗ 2 3t 2 3t 2 3t !2(t ) 0 1 = f (t )
−3 −2 −1 0 1 EE102A:Signal Processing and Linear Systems I; Win 0809 Pauly 19 Periodic signal spectrum is sampled spectrum of one cycle
F1( j!) −12!0 −8!0 −4!0 !0F1( j!)"!0 (!) 0 4!0 8!0 12!0 ! Periodic signals have sampled spectra. EE102A:Signal Processing and Linear Systems I; Win 0809 Pauly 20 Samping and Reconstruction
We can represent sampling as multiplying by an impulse train
¯
f (t) = f (t)δT (t) = ∞
n=−∞ f (nT )δ (t − nT ) In the frequency domain, the corresponds to periodization
∞
∞
1
¯ (j ω ) = 1 F (j ω ) ∗
F
δ (ω − nω0) =
F (j (ω − nω0))
T
T n=−∞
n=−∞ EE102A:Signal Processing and Linear Systems I; Win 0809 Pauly 21 Time domain:
Original Signal 3T 2T f (t ) T 0 T 2T 3T t 2T 3T t !T (t ) 1
2T 3T t × Sampling Function 3T 2T T 0 T =
Sampled Signal 3T 2T f (t )!T (t ) T 0 EE102A:Signal Processing and Linear Systems I; Win 0809 Pauly T 22 Frequency domain:
F ( j!)
−!0 −2!B 0 2!B ∗ −!0 !
1
!" (")
T0 0 ! = 1
F ( j(! + !0))
T 1
F ( j!)
T −!0 lowpass
ﬁlter −2!B 0 ¯
F (!)
1
F ( j(! − !0))
T 2!B ! We can perfectly recover F (j ω ) (and f (t)) if the sampling rate
the Nyquist rate.
EE102A:Signal Processing and Linear Systems I; Win 0809 Pauly 1
T > 2B , 23 As the sampling frequency ω0 decreases (sampling period T increases) the
spectral replicas get closer:
2!B −2!B
−!0 −2!0 −3!0 −2!0 0 −!0 −!0 0 0 !0 !0 !0 ! 2!0 ! 2!0 3!0 ! Eventually the replicas overlap, and F (j ω ) cannot be recovered.
EE102A:Signal Processing and Linear Systems I; Win 0809 Pauly 24 To recover f (t) we use an ideal lowpass ﬁlter with 2B = 1/T , sampling at
the Nyquist rate
T rect !
4"B 1
F (!)
T
−!0 −!0/2 !0 ! !0/2
2!B 0 −2!B The lowpass ﬁlter has the Fourier transform
H (j ω ) = T rect ω
4π B which has the impulse response
h(t...
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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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