Lec3 - IEG 4190 Lecture Notes 3; Lecturer: Jianzhuang Liu...

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1 IEG 4190 Lecture Notes 3; Lecturer: Jianzhuang Liu Outline: 1. Sampling and Reconstruction 2. Discrete-time Sequence 3. 2-D Sampling 4. 2-D DFS and Periodic Convolution 5. 2-D DFT and Circular Convolution 6. Exercises 7. Discrete Cosine Transform 8. Hadamard Transform 9. Orthogonal Transform Summary
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2 Sampling and Reconstruction Model of sampling – impulse modulation Let ) ( t x be a continuous time function and let ) ( t s be a uniform impulse train of period T , = n nT t t s ) ( ) ( δ The sampled time function is = × = n s nT t t x t s t x t x ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( nT t nT x nT t t x t x n n s = = Multiplication of time functions is called modulation. Therefore, multiplication by an impulse train is called impulse modulation.
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3 Sampling and Reconstruction, cont’d Therefore, we have = × = n s nT t nT x t s t x t x ) ( ) ( ) ( ) ( ) ( δ () s x t s x t
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4 Sampling and Reconstruction, cont’d The condition for recovering ) ( t x from ) ( t x s is more readily seen in the frequency domain. The CTFT of a sampled signal: ) ( ) ( ⎯→ X t x F 22 () ( ) ( ) ( ) F nn n st t nT S TT π δδ =− →Ω = ∑∑ 1 2 F ss xt x t s t X X S = = = n s F n s T n X T X nT t nT x t x ) 2 ( 1 ) ( ) ( ) ( ) ( δ
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5 Sampling and Reconstruction, cont’d () s x t s x t S S X X w w −Ω 2 T π 2 T 2 T 2 T 2 T 2 T 4 T 4 T 2 T 2 T 4 T 4 T 2 T
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6 Sampling and Reconstruction, cont’d No overlap of lobes occurs for W s W W T T > = > 2 2 2 π Thus, if a band-limited signal, i.e., W for X > = 0 ) ( is sampled with sampling frequency W s > 2 , then in principle the signal can be recovered perfectly from the samples. This is called the Sampling Theorem , and W 2 is called the Nyquist sampling rate .
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7 Sampling and Reconstruction, cont’d Reconstruction in frequency domain: ) ( ) ( ) ( = H X X s 2 T π 4 T 4 T 2 T w w w −Ω () S X H X H S xt
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8 Sampling and Reconstruction, cont’d In time domain: () ( ) ( ) ( ) ( ) ( ) ( ) ( ) s n nn xt x t ht xnT t nT ht xnT ht xnTht nT δ ⎛⎞ =∗ = −∗ ⎜⎟ ⎝⎠ = ∑∑ For the ideal lowpass filter ) ( H with cutoff frequency c , ) ( t h is given by s inc ( ) cc t T π = ( ) s [ ] n tn T xnTT ππ Ω− = H C C −Ω T sin( ) sinc( ) x x x =
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9 Sampling and Reconstruction, cont’d () xt s sinc( ) t t 1 2 3 -1 1 ( ) s inc [ ] cc n tn T xnTT ππ Ω− =
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10 Discrete-time Sequence The sampling function: ) ( ) ( ) ( nT t nT x t x n c s = −∞ = δ (previous ( )) (c=continuous-time) x t (discrete-time)
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11 Discrete-time Sequence, cont’d Different sampling rate
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12 Discrete-time Sequence, cont’d ) ( ) ( ) ( nT t nT x t x n c s = −∞ = δ It’s Fourier transform: 12 () ( ) sc n n XX TT π =−∞ Ω= Ω− (see slide 4) Since j nT tn T e − Ω , we also have ( ) jn T n Xx n T e −Ω Discrete time Fourier transform of xn : ( ) ( ) c nn n ex n T e ω −− == ∑∑ Hence, ( ) ( ) n n X ωπ
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13 Discrete-time Sequence, cont’d 12 1 2 () ( ) , ( ) ( ) sc c nn XXX X TT T T T π ωπ ω ∞∞ =−∞ Ω= Ω− = ∑∑ c xt s xn c X
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Lec3 - IEG 4190 Lecture Notes 3; Lecturer: Jianzhuang Liu...

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