Xử lý tín hiệu số 4.2 DiscreteTime Systems

1 Digital Signal Processing DiscreteTime Systems Dr. Dung Trung Vo Telecommunication Divisions Department of Electrical and Electronics September, 2013 InputOutput Rules  Discretetime system: is a processor that transforms an input sequence of discretetime samples x(n) into an output sequence of samples y(n):  Samplebysample processing:  Block processing methods:  Functional mapping:  Linear systems: this mapping becomes a linear transformation x0, x1, x2, x3,..., xn,...Hy0, y1, y2, y3,..., yn,... y  Hx y  Hx CuuDuongThanCong.com https:fb.comtailieudientucntt2  Example 1: y(n)= 2x(n), samplebysample processing  Example 2: y(n)= 2x(n)+3x(n − 1)+4x(n − 2), block processing  Example 3: y(n)= x(2n) InputOutput Rules x0, x1, x2, x3,...,H2x0,2x1,2x2,2x3,..., x0, x1, x2, x3, x4, x5, x6,...Hx0, x2, x4, x6,...  Linear system: has the property that the output signal due to a linear combination of two or more input signals can be obtained by forming the same linear combination of the individual outputs. if y1(n) and y2(n) are the outputs due to the inputs x1(n) and x2(n), then the output due to the linear combination of inputs is given by the linear combination of outputs  Testing linearity: Linearity and Time Invariance x(n)  a1x1(n)  a2x2(n) y(n)  a1y1(n)  a2 y2(n) CuuDuongThanCong.com https:fb.comtailieudientucntt3  Example 4: Test the linearity of the discretetime systems defined by and Linearity and Time Invariance y(n)  2x(n)3 y(n)  x2(n)  Time Invariance system: is a system that remains unchanged over time. This implies that if an input is applied to the system today causing a certain output to be produced, then the same output will also be produced tomorrow if the same input is applied  Right translation: Testing time invariance: delaying the input causes the output to be delayed by the same amount: Linearity and Time Invariance CuuDuongThanCong.com https:fb.comtailieudientucntt4  Example 5: Test the time invariance of the discretetime systems defined by and Linearity and Time Invariance y(n)  nx(n) y(n)  x(2n)  Impulse response of an LTI system: Linear timeinvariant systems are characterized uniquely by their impulse response sequence h(n), which is defined as the response of the system to a unit impulse δ(n).  Transform function:  Samplebysample processing: Impulse Response (n) Hh(n)     0, 0 1, 0 ( ) if n if n  n 1,0,0,0,...Hh0,h1,h2,h3,... CuuDuongThanCong.com https:fb.comtailieudientucntt5  Delayed impulse responses: time invariance implies Impulse Response (n  D) Hh(n  D)  Impulse response of an LTI system: linearity implies  Arbitrary input: {x(0), x(1), x(2), . . . } can be thought of as the linear combination of shifted and weighted unit impulses:  Its output: Impulse Response (n)(n 1)(n 2) Hh(n) h(n 1) h(n 2) y(n)  x(0)h(n) x(1)h(n 1) x(2)h(n 2)... x(n)  x(0)(n) x(1)(n 1) x(2)(n 2)... CuuDuongThanCong.com https:fb.comtailieudientucntt6 LTI form of convolution: Summation could extend over negative values of m, depending on the input signal.  Direct form of convolution: Impulse Response   m y(n) x(m)h(n m)   m y(n) h(m)x(n m)  FIR: has impulse response h(n) that extends only over a finite time interval, say 0 ≤ n ≤ M, and is identically zero beyond that  Filter order: M  Filter length: length of the impulse response vector h = h0 , h1, . . . , hM  Impulse response coefficients: h = h0 , h1, . . . , hM are referred to by various names, such as filter coefficients, filter weights, or filter taps  FIR filtering equation: FIR and IIR Filters    M m y n h m x n m 0 ( ) ( ) ( ) L  M 1 h h0,h1,h2,h3,...,hM ,0,0,0,... CuuDuongThanCong.com https:fb.comtailieudientucntt7  IIR: has an impulse response h(n) of infinite duration, defined over the infinite interval 0 ≤ n < ∞  IIR filtering equation:  Constant coefficient linear difference equations: This IO equation is not computationally feasible. Therefore, we must restrict our attention to a subclass of IIR filters, namely, those filter coefficients are coupled to each other through constant coefficient linear difference equations:  Difference equation for y(n): FIR and IIR Filters    0 ( ) ( ) ( ) m y n h m x n m         L i i M i h n aih n i b n i 1 1 ( ) ( )  ( )         L i i M i y n ai y n i b x n i 1 1 ( ) ( ) ( )  Example 6: Determine the impulse response h of the following FIR filters FIR and IIR Filters ) ( ) ( ) ( 4) ) ( ) 2 ( ) 3 ( 1) 5 ( 2) 2 ( 3)           b y n x n x n a y n x n x n x n x n CuuDuongThanCong.com https:fb.comtailieudientucntt8  Example 7: Determine the IO difference equation of an IIR filter whose impulse response coefficients h(n) are coupled to each other by the difference equation FIR and IIR Filters h(n)  h(n 1)(n)  Example 8: Suppose the filter coefficients h(n) satisfy the difference equation where a is a constant. Determine the IO difference equation relating a general input signal x(n) to the corresponding output y(n). FIR and IIR Filters h(n)  ah(n 1)(n) CuuDuongThanCong.com https:fb.comtailieudientucntt9  Causal signal: A causal or rightsided signal x(n) exists only for n ≥ 0 and vanishes for all negative times n ≤ −1  Anticausal signal: An anticausal or leftsided signal exists only for n ≤ −1 and vanishes for all n ≥ 0.  Mixed signal: A mixed or doublesided signal has both a leftsided and a rightsided part LTI systems can also be classified in terms of their causality properties depending on whether their impulse response h(n) is causal, anticausal, or mixed Causality  Description: a system is stable if the output remains bounded by some bound |y(n)| ≤ B if its input is bounded, say |x(n)| ≤ A.  Necessary and sufficient condition: Stability is absolutely essential in hardware or software implementations of LTI systems because it guarantees that the numerical operations required for computing the IO convolution sums or the equivalent difference equations remain well behaved and never grow beyond bounds. The concepts of stability and causality are logically independent, but are not always compatible with each other Stability     n h(n) CuuDuongThanCong.com https:fb.comtailieudientucntt10  Example 9: Consider the following four examples of h(n): Stability  Homework: provided in class Stability CuuDuongThanCong.com https:fb.comtailieudientucntt

Input/Output Rules Digital Signal Processing  Discrete-time system: is a processor that transforms an input sequence of discrete-time samples x(n) into an output sequence of samples y(n): Discrete-Time Systems  Sample-by-sample processing: Dr Dung Trung Vo Telecommunication Divisions Department of Electrical and Electronics H x0 , x1 , x2 , x3 , , xn ,   y0 , y1 , y2 , y3 , , yn ,   Block processing methods:  Functional mapping: September, 2013 Input/Output Rules  Example 1: y(n)= 2x(n), sample-by-sample processing H x0 , x1 , x2 , x3 , ,  2 x0 ,2 x1 ,2 x2 ,2 x3 , ,  Example 2: y(n)= 2x(n)+3x(n − 1)+4x(n − 2), block processing y  H x   Linear systems: this mapping becomes a linear transformation y  Hx Linearity and Time Invariance  Linear system: has the property that the output signal due to a linear combination of two or more input signals can be obtained by forming the same linear combination of the individual outputs if y1(n) and y2(n) are the outputs due to the inputs x1(n) and x2(n), then the output due to the linear combination of inputs x(n)  a1 x1 (n)  a2 x2 (n) is given by the linear combination of outputs y (n)  a1 y1 (n)  a2 y2 (n)  Testing linearity:  Example 3: y(n)= x(2n) H x0 , x1 , x2 , x3 , x4 , x5 , x6 ,   x0 , x2 , x4 , x6 ,  CuuDuongThanCong.com https://fb.com/tailieudientucntt Linearity and Time Invariance Linearity and Time Invariance  Example 3.2.1: Test the linearity of the discrete-time systems defined by  Time Invariance system: is a system that remains unchanged over time This implies that if an input is applied to the system today causing a certain output to be produced, then the same output will also be produced tomorrow if the same input is applied y (n)  x(n)  and y (n)  x (n)  Solution: the output due to the linear combination will be y (n)  x(n)   2a1 x1 (n)  a2 x2 (n)    Right translation: is not equal to the linear combination of each input a1 y1 (n)  a2 y2 (n)  a1 (2 x1 (n)  3)  a2 (2 x2 (n)  3) Testing time invariance: delaying the input causes the output to be delayed by the same amount: So it is not a linear system Similarly, for the quadratic system y ( n)  x ( n) a x (n)  a x (n)  a1 x1 (n)  a2 x2 (n)  1 2 2 Linearity and Time Invariance Impulse Response  Example 3.2.2: Test the time invariance of the discrete-time systems defined by y (n)  nx(n) and y (n)  x(2n)  Impulse response of an LTI system: Linear time-invariant systems are characterized uniquely by their impulse response sequence h(n), which is defined as the response of the system to a unit impulse δ(n)  Solution:  Delaying the input signal y D (n)  nxD (n)  nx(n  D) 1, if 0, if  ( n)   n0 n0  Delaying the output signal y (n  D)  (n  D) x(n  D)  nx(n  D)  y D (n) Thus, the system is not time-invariant Similarly, for system y(n)= x(2n) y D (n)  nxD (2n)  nx(2n  D) y (n  D)  x(2(n  D))  x(2n  D)  y D (n) Thus, the downsampler is not time-invariant  Transform function: H  (n)  h ( n)  Sample-by-sample processing: H 1,0,0,0,   h0 , h1 , h2 , h3 ,  CuuDuongThanCong.com https://fb.com/tailieudientucntt Impulse Response  Delayed impulse responses: time invariance implies H  (n  D)  h( n  D ) Impulse Response  Impulse response of an LTI system: linearity implies H  (n)   (n  1)   (n  2)  h(n)  h(n  1)  h(n  2)  Arbitrary input: {x(0), x(1), x(2), } can be thought of as the linear combination of shifted and weighted unit impulses: x(n)  x(0) (n)  x(1) (n  1)  x(2) (n  2)   Its output: y (n)  x(0)h(n)  x(1)h(n  1)  x(2)h(n  2)  Impulse Response FIR and IIR Filters  FIR: has impulse response h(n) that extends only over a finite time interval, say ≤ n ≤ M, and is identically zero beyond that h0 , h1 , h2 , h3 , , hM ,0,0,0,   Filter order: M  Filter length: length of the impulse response vector h = [h0 , h1, , hM] LTI form of convolution: y ( n )   x ( m ) h ( n  m) m Summation could extend over negative values of m, depending on the input signal  Direct form of convolution: y ( n )   h ( m ) x ( n  m) Lh  M   Impulse response coefficients: h = [h0 , h1, , hM] are referred to by various names, such as filter coefficients, filter weights, or filter taps  FIR filtering equation: M y ( n )   h( m) x ( n  m ) m 0 m CuuDuongThanCong.com https://fb.com/tailieudientucntt FIR and IIR Filters  IIR: has an impulse response h(n) of infinite duration, defined over the infinite interval ≤ n < ∞  IIR filtering equation: FIR and IIR Filters  Example 1: Determine the impulse response h of the following FIR filters a) y (n)  x(n)  3x(n  1)  x(n  2)  x(n  3) b) y (n)  x(n)  x(n  4)  y ( n )   h ( m ) x ( n  m) m 0  Constant coefficient linear difference equations: This I/O equation is not computationally feasible Therefore, we must restrict our attention to a subclass of IIR filters, namely, those filter coefficients are coupled to each other through constant coefficient linear difference equations: M L i 1 i 1 h(n)   h(n  i )   bi (n  i) Solution: impulse response coefficients a) h  h0 , h1 , h2 , h3   2,3,5,2 b) h  h0 , h1 , h2 , h3 , h4   1,0,0,0,1 Unit impulse as input, x(n)= δ(n): a ) h(n)  2 (n)  3 (n  1)  5 (n  2)  2 (n  3) b) h(n)   (n)   (n  4)  Difference equation for y(n): M L i 1 i 1 y (n)   y (n  i )   bi x(n  i) FIR and IIR Filters  Example 2: Determine the I/O difference equation of an IIR filter whose impulse response coefficients h(n) are coupled to each other by the difference equation h(n)  h(n  1)   (n)  Solution: Setting n = 0, we have h(0)= h(−1)+δ(0)= h(−1)+1 Assuming causal initial conditions, h(−1)= 0, we find h(0)= For n > 0, the delta function vanishes, δ(n)= 0, and therefore, the difference equation reads h(n)= h(n−1) 1, if h ( n)  u ( n )   0, if n0 n0  Convolutional I/O equation:   m 0 m 0 y ( n )   h ( m) x ( n  m)   x ( n  m)  Recursive difference equation y (n)  x(n)  x(n  1)  x(n  2)  x(n  3)  FIR and IIR Filters  Example 3: Suppose the filter coefficients h(n) satisfy the difference equation h(n)  ah(n  1)   (n) where a is a constant Determine the I/O difference equation relating a general input signal x(n) to the corresponding output y(n) Solution: filter coefficients a n , if h ( n)  a n u ( n)    0, if n0 n0 I/O difference equation y (n)  x(n)  ax(n  1)  a x(n  2)  a x(n  3)    y (n)  x(n)  a x(n  1)  ax(n  2)  a x(n  3)  y (n)  ay (n  1)  x(n) y (n)  y (n  1)  x(n) CuuDuongThanCong.com https://fb.com/tailieudientucntt Causality Stability  Causal signal: A causal or right-sided signal x(n) exists only for n ≥ and vanishes for all negative times n ≤ −1  Description: a system is stable if the output remains bounded by some bound |y(n)| ≤ B if its input is bounded, say |x(n)| ≤ A  Anti-causal signal: An anticausal or left-sided signal exists only for n ≤ −1 and vanishes for all n ≥  Necessary and sufficient condition:  Mixed signal: A mixed or double-sided signal has both a left-sided and a rightsided part   h( n)   n   Stability is absolutely essential in hardware or software implementations of LTI systems because it guarantees that the numerical operations required for computing the I/O convolution sums or the equivalent difference equations remain well behaved and never grow beyond bounds The concepts of stability and causality are logically independent, but are not always compatible with each other LTI systems can also be classified in terms of their causality properties depending on whether their impulse response h(n) is causal, anticausal, or mixed Stability  Example 4: Consider the following four examples of h(n): Stability  Homework: 3.1, 3.2, 3.3, 3.4, 3.6  Solution:   n   n     n   n  1    h(n)   0.5  h(n)   0.5  h(n)   n    n n n   0.5    m   m 1  n 0   n   n  1   h(n)     0.5 n m 1 n  0.5   0.5 CuuDuongThanCong.com https://fb.com/tailieudientucntt ... ≤ A  Anti-causal signal: An anticausal or left-sided signal exists only for n ≤ −1 and vanishes for all n ≥  Necessary and sufficient condition:  Mixed signal: A mixed or double-sided signal... system is not time-invariant Similarly, for system y(n)= x(2n) y D (n)  nxD (2n)  nx(2n  D) y (n  D)  x(2(n  D))  x(2n  D)  y D (n) Thus, the downsampler is not time-invariant  Transform... (n) Thus, the downsampler is not time-invariant  Transform function: H  (n)  h ( n)  Sample-by-sample processing: H 1,0,0,0,   h0 , h1 , h2 , h3 ,  CuuDuongThanCong.com https://fb.com/tailieudientucntt