Créer une présentation
Télécharger la présentation

Télécharger la présentation
## Linear Constant-coefficient Difference Equations

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Linear Constant-coefficient Difference Equations**for alln • An important subclass of linear time-invariant systems consist of those system for which the input x[n] and output y[n] satisfy an Nth-order linear constant-coefficient difference equation. • A general form is shown above.**+**+ + + + + + + x[n] y[n] b0 TD TD TD TD TD TD bM b2 b1 a2 aN a1 y[n-1] x[n-1] x[n-2] y[n-2] y[n-N] x[n-M] Signal Flow Graph of the Difference Equation • Assume that a0 = 1. Let TD denote one-sample delay.**Difference Equation: FIR system**• The assumption a0 = 1 can be always achieved by dividing all the coefficients by a0if a00. • The difference equation characterizes a recursive way of obtaining the output y[n] from the input x[n]. • When ak = 0 for k = 1 … N, the difference equation degenerates to a FIR system. • The output consists of a linear combination of finite inputs.**Difference equation: IIR System**• When bmare not all zerosfor m = 1 … M, the difference equation degenerates to • This causes an IIR system • The effect of an impulse response sequence applied to the input keeps on circulating around the feedback loops indefinitely.**Example**• Accumulator**Example (continue)**• Moving average system when M1=0: • The impulse response is h[n] = u[n] u[nM2 1] • Also, note that The term y[n] y[n1] suggests the implementation can be cascaded with an accumulator.**+**+ + + b TD TD TD x[n] y[n] b b b x[n-1] x[n-2] x[n-M] Moving Average System • Hence, there are at least two difference equation representations of the moving average system. First, where b = 1/ (M2+1) and TD denotes one-sample delay**Moving Average System (continue)**• Second, • The first representation is FIR, and the second is IIR.**Solution of Difference Equation**• Just as differential equations for continuous-time systems, a linear constant-coefficient difference equation for discrete-time systems does not provide a unique solution if no additional constraints are provided. • Solution: y[n] = yp[n] + yh[n] • yh[n]: homogeneous solution obtained by setting all the inputs as zeros. • yh[n]: a particular solution satisfying the difference equation.**Solution of Difference Equation**• Additional constraints: consider the N auxiliary conditions that y[-1], y[-2], …, y[-N] are given. • The other values of y[n] (n0) can be generated by when x[n] is available, y[1], y[2], … y[n], … can be computed recursively. • To generate values of y[n] for n<N recursively,**Example of the Solution**• Consider the difference equation y[n] = ay[n-1] + x[n]. • Assume the input is x[n] =K [n], and the auxiliary condition is y[1] = c. • Hence, y[0] = ac+K, y[1] = a y[0]+0 = a2c+aK, … • Recursively, we found that y[n] = an+1c+anK, forn0. • For n<1, y[-2] = a1(y[1]x[1] ) = a1c, y[2] = a1 y[1] = a2 c, …, and y[n] = an+1c for n<1. • Hence, the solution is y[n] = an+1c+Kanu[n],**Example of the Solution (continue)**• The recursively-implemented system for finding the solution is non-causal. • The solution system is non-linear: • When K=0, i.e., the input is zero, the solution (system response) y[n] = an+1c. • Since a linear system requires that the output be zero for all time when the input is zero for all time. • The solution system is not shift invariant: • when input were shifted by n0 samples, x1[n] =K [n - n0], the output is y1[n] = an+1c+Kann0u[n - n0].**LTI solution**• Our principal interest in the text is in systems that are linear and time invariant. • How to make the recursively-implemented solution system be LTI? • Initial-rest condition: • If the input x[n] is zero for n less than some time n0, the output y[n] is also zero for n less than n0. • The previous example does not satisfy this condition since x[n]= 0 for n<0 but y[1] = c. • Property: If the initial-rest condition is satisfied, then the system will be LTI and causal.**Frequency-Domain Representation of Discrete-time Signals and**Systems • Eigen function of a LTI system • When apply an eigenfunction as input, the output is the same function multiplied by a constant. • x[n] = ejwn is the eigenfunction of all LTI systems. • Let h[n] be the impulse response of an LTI system, when ejwn is applied as the input,**Eigenfunction of LTI**• Let we have • consequently, ejwn is the eigenfunction of the system, and the associated eigenvalue is H(ejw). • We call H(ejw) the LTI system’s frequency response that consists of the real and imaginary parts, H(ejw)=HR(ejw)+jHI(ejw), or in terms of magnitude and phase,**Example of Frequency Response**• Frequency response of the ideal delay system, y[n] =x[n nd], • If we consider x[n] = ejwn as input, then Hence, the frequency response is • The magnitude and phase are**Linear Combination**• When a signal can be represented as a linear combination of complex exponentials: By the principle of superposition, the output is • Thus, we can find the output if we know the frequency response of the system.**Example of Linear Combination**• Sinusoidal responses of LTI systems: • The response of x1[n] and x2[n] are • If h[n] is real, it can be shown that H(e-jw0) = H*(ejw0), the total response y[n]=y1[n]+ y2[n] is**Difference to Continuous-time System Response**• For a continuous-time system, the frequency response is not necessarily to be periodic. • However, for a discrete-time system, the frequency response is always periodic with period 2, since • Because H(ejw) is periodic with period 2, we need only specify H(ejw) over an interval of length 2, eg., [0,2] or [,]. For consistency, we choose the interval [,]. • The inherent periodicity defines the frequency response everywhere outside the chosen interval.**Ideal Frequency-selective Filters**• The “low frequencies” are frequencies close to zero, while the “high frequencies” are those close to . • Since that the frequencies differing by an integer multiple of 2 are indistinguishable, the “low frequency” are those that are close to an even multiple of , while the “high frequencies” are those close to an odd multiple of . • Ideal frequency-selective filters: • An important class of linear-invariant systems includes those systems for which the frequency response is unity over a certain range of frequencies and is zero at the remaining frequencies.**Frequency Response of the Moving-average System**• The impulse response of the moving-average system is • Therefore, the frequency response is • By noting that the following formula holds:**Frequency Response of the Moving-average System (continue)**(magnitude and phase)**Frequency Response of the Moving-average System (continue)**M1= 0 and M2 = 4 Amplitude response Phase response 2w**Suddenly Applied Complex Exponential Inputs**• In practice, we may not apply the complex exponential inputs ejwn to a system, but the more practical-appearing inputs of the form x[n] = ejwn u[n] • i.e., complex exponentials that are suddenly applied at an arbitrary time, which for convenience we choose n=0. • Consider its output to a causal LTI system:**Suddenly Applied Complex Exponential Inputs (continue)**• We consider the output for n 0. • Hence, the output can be written as y[n] = yss[n] + yt[n], where Steady-state response Transient response**Suddenly Applied Complex Exponential Inputs (continue)**• If h[n] = 0 except for 0 n M(i.e., a FIR system),then the transient response yt[n] = 0 for n+1 > M. That is, the transient response becomes zero since the time n = M. For n M, only the steady-state response exists. • For infinite-duration impulse response (i.e., IIR) • For stable system, Qn must become increasingly smaller as n , and so is the transient response.**Suddenly Applied Complex Exponential Inputs (continue)**Illustration for the FIR case by convolution**Suddenly Applied Complex Exponential Inputs (continue)**Illustration for the IIR case by convolution**Representation of Sequences by Fourier Transforms**• Fourier Representation: representing a signal by complex exponentials. • A signal x[n] is represented as the Fourier integral of the complex exponentials in the range of frequencies [,]. • The weight X(ejw) of the frequency applied in the integral can be determined by the input signal x[n], and X(ejw) reveals how much of each frequency is required to synthesize x[n]. Inverse Fourier transform Fourier transform (or forward Fourier transform)**Representation of Sequences by Fourier Transforms (continue)**• The phase X(ejw) is not uniquely specified since any integer multiple of 2 may be added to X(ejw) at any value of w without affecting the result. • Denote ARG[X(ejw)] to be the phase value in [,]. • Since the frequency response of a LTI system is the Fourier transform of the impulse response, the impulse response can be obtained from the frequency response by applying the inverse Fourier transform integral:**Existence of Fourier Transform Pairs**• Whey they are transform pairs? Consider the integral**Conditions for the Existence of Fourier Transform Pairs**• Conditions for the existence of Fourier transform pairs of a signal: • Absolutely summable • Mean-square convergence: In other words, the error |X(ejw) XM(ejw) | may not approach for each w, but the total “energh” in the error does.**Conditions for the Existence of Fourier Transform Pairs**(continue) • Still some other cases that are neither absolutely summable nor mean-square convergence, the Fourier transform still exist: • Eg., Fourier transform of a constant, x[n] = 1 for all n, is an impulse train: • The impulse of the continuous case is a “infinite heigh, zero width, and unit area” function. If some properties are defined for the impulse function, then the Fourier transform pair involving impulses can be well defined too.**Conditions for the Existence of Fourier Transform Pairs**(continue) • Properties of continuous impulse function: • Eg., consider a sequence whose Fourier transform is the periodic impulse train then the sequence is a complex exponential sequence (note that extends only over one period, from, we need include the term)**Symmetry Property of the Fourier Transform**• Conjugate-symmetric sequence: xe[n] = xe*[n] • If a real sequence is conjugate symmetric, then it is called an even sequence satisfying xe[n] = xe[n]. • Conjugate-asymmetric sequence: xo[n] = xo*[n] • If a real sequence is conjugate antisymmetric, then it is called an odd sequence satisfying x0[n] = x0[n]. • Any sequence can be represented as a sum of a conjugate-symmetric and asymmetric sequences, x[n] =xe[n] + xo[n], where xe[n] = (1/2)(x[n]+ x*[n]) and xo[n] = (1/2)(x[n] x*[n]).**Symmetry Property of the Fourier Transform (continue)**• Similarly, a Fourier transform can be decomposed into a sum of conjugate-symmetric and anti-symmetric parts: X(ejw) = Xe(ejw) + Xo(ejw) , where Xe(ejw) = (1/2)[X(ejw) + X*(ejw)] and Xo(ejw) = (1/2)[X(ejw) X*(ejw)]**Symmetry Property of the Fourier Transform (continue)**• Fourier Transform Pairs (if x[n] X(ejw)) • x*[n] X*(e jw) • x*[n] X*(ejw) • Re{x[n]} Xe(ejw) (conjugate-symmetry part ofX(ejw)) • jIm{x[n]} Xo(ejw) (conjugate anti-symmetry part ofX(ejw)) • xe[n] (conjugate-symmetry part ofx[n]) XR(ejw) = Re{X(ejw)} • xo[n] (conjugate anti-symmetry part ofx[n]) jXI(ejw) = jIm{X(ejw)}**Symmetry Property of the Fourier Transform (continue)**• Fourier Transform Pairs (if x[n] X(ejw)) • Any realxe[n] X(ejw) = X*(ejw) (Fourier transform is conjugate symmetric) • Any realxe[n] XR(ejw) = XR(ejw) (real part is even) • Any realxe[n] XI(ejw) = XI(ejw) (imaginary part is odd) • Any realxe[n] |XR(ejw)| = |XR(ejw)| (magnitude is even) • Any realxe[n] XR(ejw)= XR(ejw) (phase is odd) • xo[n] (even part of realx[n]) XR(ejw) • xo[n] (odd part of realx[n]) jXI(ejw)**Example of Symmetry Properties**• The Fourier transform of the real sequence x[n] = anu[n] for a < 1 is • Its magnitude is an even function, and phase is odd.**Fourier Transform Theorems**• Linearity x1[n] X1(ejw), x2[n] X2(ejw) implies that a1x1[n] + a2x2[n] a1X1(ejw) + a2X2(ejw) • Time shifting x[n] X(ejw) implies that**Fourier Transform Theorems (continue)**• Frequency shifting x[n] X(ejw) implies that • Time reversal x[n] X(ejw) If the sequence is time reversed, then x[n] X(ejw) Furthermore, ifx[n] is real then x[n] X*(ejw), since X(ejw) is conjugate symmetric.**Fourier Transform Theorems (continue)**• Differentiation in frequency x[n] X(ejw) implies that • Parseval’s theorem x[n] X(ejw) implies that**Fourier Transform Theorems (continue)**• The convolution theorem x[n] X(ejw) and h[n] H(ejw), and ify[n] = x[n] h[n], then Y(ejw) = X(ejw)H(ejw) • The modulation or windowing theorem x[n] X(ejw) and w[n] W(ejw), and ify[n] = x[n]w[n], then a periodic convolution