How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches Difference Equations (an LTI system) x[n]: input, y[n]: output That is, building a system that maes use of the current and previous inputs x[n], x[n-], x[n- 2]. as well as the previous outputs y[n-], y[n-2], Discrete Fourier Transform (in short, DFT) Remember we have introduced three inds of Fourier transforms. However, what we are able to deal with in the discrete-time domain is usually a finiteduration signal. DFT is the final (fourth) Fourier transform, where its input is a discrete-time finite-duration signal. Details will be introduced in later courses
DTFT Theorems Reviews (From Oppenheim) Linearity x [n] X (e jw ), x 2 [n] X 2 (e jw ) implies that a x [n] + a 2 x 2 [n] a X (e jw ) + a 2 X 2 (e jw ) Time shifting x[n] X(e jw ) implies that x jwn jw d n n e X e d
DTFT Theorems (continue) Frequency shifting x[n] X(e jw ) implies that Time reversal e jw n x j w n X e 0 w 0 x[n] X(e jw ) If the sequence is time reversed, then x[n] X(e jw )
DTFT Theorems (continue) Differentiation in frequency implies that x[n] X(e jw ) Parseval s theorem x[n] X(e jw ) implies that E nx n n x j dx e dw jw 2 jw n X e 2 2 dw
DTFT Theorems (continue) The convolution theorem x[n] X(e jw ) and h[n] H(e jw ), and if y[n] = x[n] h[n], then Y(e jw ) = X(e jw )H(e jw ) The modulation or windowing theorem x[n] X(e jw ) and w[n] W(e jw ), and if y[n] = x[n]w[n], then Y jw j j w e X e W e 2 d Periodic convolution, Y(e jw )= Y(e j(w+2) )
DTFT Theorems (continue) Hence, unlie continuous Fourier transform, DTFT s time and frequency domains are not that analogous in DTFT. For example, convolution in the time domain implies multiplication in the frequency domain. But multiplication in the time domain implies periodic-convolution in the frequency domain.
DTFT Pairs n n n e 0 jw n 0 ( n ) 2 w 2
DTFT Pairs (continue) Real Exponentials n a u n ( a ) ae u n w 2 jw e n n a u n ( a ) jw jw ae 2
n DTFT Pairs (continue) sin(ax)/x : called sync function sin(ax)/sin(x) : called Dirichlet Kernel r sin w n p u n ( r ) sin w 2 r cos w e r e p p jw 2 j 2w sin n wn c X e jw 0 w c w w w c x n 0 n M sin w M / 2 0 otherw ise sin w / 2 e jw M /2
DTFT Pairs (finally) jw n 0 e 2 w w 2 0 j j w n e w w e w w cos 2 2 0 0 0
Example: Determining a Fourier Transform Pairwise Formulas Suppose we wish to find the Fourier transform of x[n] = a n u[n-5]. x x x 2 n jw n a u n X e Thus, n x n 5 X 2 ae jw j 5 w jw e e X e 5 jw n a x 2 n, so X e jw jw a 5 e ae e j 5 w j 5 w ae jw
Another Example Determining the impulse response for a difference equation y[n](/2) y[n] = x[n] (/4)x[n] To find the impulse response, we set x[n] = [n]. Then the above equation becomes h[n](/2) h[n] = [n] (/4)[n] Applying DTFT, we obtain H(e jw ) (/2)e -jw H(e jw ) = (/4) e -jw So H(e jw ) = ( (/4) e -jw ) / ( (/2) e -jw )
continue Then, from the pairwise table, we now a n u n ( a ) ae jw thus, (/2) n u[n] / ( (/2) e -jw ) By the shifting property, in the time domain, the second term of H(e jw ) is (/4)(/2) n u[n] (/4) e -jw / ( (/2) e -jw ) Hence, h[n] = (/2) n u[n] (/4)(/2) n u[n]
Symmetry Property of DTFT Conjugate-symmetric sequence: x e [n] = x e *[ n] If a real sequence is conjugate symmetric, then it is called an even sequence satisfying x e [n] = x e [ n]. Conjugate-asymmetric sequence: x o [n] = x o *[ n] If a real sequence is conjugate antisymmetric, then it is called an odd sequence satisfying x 0 [n] = x 0 [n]. Any sequence can be represented as a sum of a conjugate-symmetric and asymmetric sequences, x[n] = x e [n] + x o [n], where x e [n] = (/2)(x[n]+ x*[n]) and x o [n] = (/2)(x[n] x*[n]).
Symmetry Property of DTFT (continue) Similarly, a Fourier transform can be decomposed into a sum of conjugate-symmetric and antisymmetric parts: where and X(e jw ) = X e (e jw ) + X o (e jw ), X e (e jw ) = (/2)[X(e jw ) + X*(e jw )] X o (e jw ) = (/2)[X(e jw ) X*(e jw )]
Symmetry Property of DTFT (continue) Fourier Transform Pairs (if x[n] X(e jw )) x*[n] X*(e jw ) x*[n] X*(e jw ) Re{x[n]} X e (e jw ) (conjugate-symmetry part of X(e jw )) jim{x[n]} X o (e jw ) (conjugate anti-symmetry part of X(e jw )) x e [n] (conjugate-symmetry part of x[n]) X R (e jw ) = Re{X(e jw )} x o [n] (conjugate anti-symmetry part of x[n]) jx I (e jw ) = jim{x(e jw )}
Symmetry Property of DTFT (continue) Fourier Transform Pairs (if x[n] X(e jw )) Any real x e [n] X(e jw ) = X*(e jw ) (Fourier transform is conjugate symmetric) Any real x e [n] X R (e jw ) = X R (e jw ) (real part is even) Any real x e [n] X I (e jw ) = X I (e jw ) (imaginary part is odd) Any real x e [n] X R (e jw ) = X R (e jw ) (magnitude is even) Any real x e [n] X R (e jw )= X R (e jw ) (phase is odd) x o [n] (even part of real x[n]) X R (e jw ) x o [n] (odd part of real x[n]) jx I (e jw )
Example of Symmetry Properties The Fourier transform of the real sequence x[n] = a n u[n] for a < is X jw e jw Its magnitude is an even function, and phase is odd. ae
Sinusoidal Responses of LTI System (application of symmetric property) We have seen that e jwn is an eigen function of any LTI system, with the corresponding eigenvalue (or frequency response) H(e jw ). What happens when feeding a real sinusoid of frequency w 0 into the LTI system with the frequency response H(e jw )? A j jw n A j x n A cos w n e e e e 0 2 2 0 0 jw n
Sinusoidal Responses of LTI System (application of symmetric property) If the impulse response h[n] is real, by the symmetry property, H(e jw 0) = H*(e jw 0) Hence, A j A j y n H e e e H e e e 2 2 0 0 R e{ } jw jw n jw jw n 0 0 0 0 A jw0 j jw0n A jw0 j jw0n H e e e H e ( e e ) 2 2 A jw0 j jw0n A jw0 j jw0n H e e e [ H e e e ] 2 2 jw j A H e e e jw n jw0 j jw0n j jw0 R e{ }, w here A H e e e e H e jw0 cos 0 A H e w n
Efficient Representation of Difference Equations We have mentioned that difference equation is a common way for realiing an LTI system (although not all LTI system is able to be implemented by difference equations) N 0 a y M n bm xn m m 0 To represent a difference equation efficiently, -transform introduced below is widely used.
Z-transform Representing a discrete-time signal (or a sequence) as a polynomial. X n n For CS students: It has the same form of the generating function for combinations in combinatorics. But are developed from different domains. In fact, the difference equation is the same as the recurrence relation in combinatorics. x n
Z-transform For a right-sided sequence (x[n]=0 for n<0), the -transform is X x n n 0 Example: what is the -transform of deltafunction? X n n 0 n n
Example of -transform for a finite sequence 5 4 3 2 2 4 6 4 2 X n n 0 2 3 4 5 N>5 x[n] 0 2 4 6 4 2 0
Z-Transform vs. DTFT Discrete-time Fourier Transform X jw e xn Hence, DTFT is equivalent to substituting =e jw into the -transform n More specifically, the -transform is a generaliation of the DTFT, where the DTFT evaluates the -transform only on the complex unit circle ( = ): e jwn n jw X x n n e
Z-Transform vs. DTFT The unit circle in the complex plane. DTFT is only evaluated on this circle.
Z-transform vs. Convolution Remember that the convolution results can be seen as the coefficients of polynomial products. Since -transform represents a sequence as a polynomial, it has the property that Time domain convolution implies Z-domain multiplication
Proof Convolution Tae the -transform on both sides:
Proof (con t) Time domain convolution implies Z-domain multiplication
Time Delay Property It can be easily shown that time delay of n 0 samples is equivalent to multiplying -n 0 in the -domain.
Z-transform Applying to Systems In the above, -transform is applied to a signal. Now, we apply it to the LTI system realied by a difference equation: Taing -transforms for both sides, we have M m m N m n x b n y a 0 0 m M m m N X b Y a 0 0
Z-transform Applying to Systems Hence Y X M m 0 N 0 b a m We call Y H ( ) A fractional form X the system function of this LTI system General Definition: the system function of an LTI system is the -transform of the output signal divided by the -transform of the input signal. m
System Function vs. Frequency Response Hence, when feeding an input signal X() to an LTI system with the system function H(), the output is the product Y()=X()H() When =e jw, we obtain the output spectrum Y(e jw )=X(e jw )H(e jw ). X() Y() (= H()X()) H()/H(e jw ) X(e jw ) Y(e jw ) (= H(e jw )X(e jw )) Z-transform Fourier transform
Z-transform and Impulse Response In addition, the convolution in time domain implies multiplication in the -domain (and frequency domain). So, x[n] h[n] Time domain convolution y[n] = x[n]h[n] X() Y() (= H()X()) H()/H(e jw ) X(e jw ) Y(e jw ) (= H(e jw )X(e jw )) Z-transform Fourier transform
Z-transform Hence, we can represent an LTI system by either h[n] (impulse response), H(e jw ) (frequency response), or H() (-transform) In particular, we usually use H() as a fractional form: M H ( ) m 0 N 0 b to represent a difference-equation LTI system. m a m
Cascading of LTI Systems Cascading of LTI systems can be described in the -domain too.
Z-transform vs. Laplace transform Remar Z-transform transfers a discrete-time signal to the -domain. It is analogous to the Laplace transform for the continuous-time signal that is transformed to the s- domain. Z-transform: for solving difference equations Laplace transform: for solving differential equations
Time Delay System Recall that time delay of n 0 samples is equivalent to convolving with (n-n 0 ) in time domain, or multiplying -n 0 in the -domain. We call - the unit-delay system
System Diagram of A Causal FIR System The signal-flow graph of a causal FIR system can be represented by -transforms: x[n] b 0 + y[n] y M n b m x n m m 0 x[n-] x[n-2] b b 2 + + x[n-m] b M +
Deconvolution Deconvolution or inverse filtering
Example Deconvolution or inverse filtering
Poles and Zeros of Fractional Form Let H() = P()/Q() be the system function of a rational form. Let us represent both P() and Q() as polynomials of (not - ) Then Poles: the roots of Q() Zeros: the roots of P()
Example of Poles and Zeros
Example and Physical Meanings Three eros: Three poles: 0, 0, 0 (or called a three-order pole at 0) Physical meanings Pole: The pole of a -transform H() are the values of for which H()=. Zero: The ero of a -transform H() are the values of for which H()=0.
Poles and eros of negative powers of
Pole-Zero Plot It is useful to display the eros and poles of H() on the complex -plane. Knowing the poles and eros is equivalent to nowing the LTI system. eros poles
Significance of the Zeros of H() for The poles a FIR system are always in the origin (or a multi-order ero). The eros of a FIR filter thus play a major role in the frequency response. Recall that DTFT is evaluated on the unitcircle on the -plane. If the eros are placed close to the unit circle, we can image that the nearby frequencies will be lowered. FIR System
Example: The L-point Running- Sum Filter The system function is We have to represent it as rational form in. From geometric series,
Example: The L-point Running- Sum Filter The eros of H() will be determined by the roots of the numerator polynomial, The poles are determined by the denominator. There are (L-)-th roots at =0 and a root at =. So, the is a common location at = for both pole and ero; they are canceled. Hence, finally, the L-point runing-sum filter has L- eros and an (L-)th-order pole at the origin.
Example: The L-point Running- For instance, L=0 Sum Filter
Example: The L-point Running- Sum Filter The magnitude response: note that there are a total of 9 ero responses from [-, )
Recall: Ideal Low-pass Filter s frequency response
Passband In the above example of 0-point running sum filter, the passband is roughly centered at w=0. It acts lie a low-pass filter. Can we move the passband to other locations by re-placing eros, so that the filter acts lie some other frequency-selection filters?
Frequency-selection Filters: Ideal High-pass Filter
Ideal Band-pass Filter
Ideal Band-stop Filter
Bandpass Filter by Placing the Zeros of an FIR system We can move the passband to a frequency away from w=0 by changing the ero locations: The obvious way is to use all but one of the roots of unity as the eros of an FIR filter. Note that by specifying poles and eros, we completely describe the LTI system. L=0
Bandpass Filter by Placing the Zeros of an FIR system Frequency magnitude response:
Bandpass Filter with Real Coefficients However, the above filter have complex coefficients. By the symmetry property of DTFT, we now that the magnitude response should be an even function if we hope the coefficients are all real.
Practical Band Pass Filtering The above examples enable us a process of manipulating (adding, moving, removing) the eros interactively by using a simulation system (eg. Matlab) for filter design. We show a filter obtained by such an interactive design below, although much better filters can be deigned by more sophisticated methods. It is a useful illustration of the power of the - transform to simplify the analysis of such problems. The underlying reason is that the -transform coverts difficult problems involving convolution and frequency response into simple algebraic ideas based multiplying and factoring polynomials.
Practical Band Pass Filtering
Digital Filter Structures We now that there could be multiple ways to realie the system function of a difference equation. Z-transform is also useful in help characterie these solutions, and find an efficient implementation structure.
Digital Filter Structures (for IIR Filter) 0 N M y n a y n b x n M n x b n v 0 ] [ ] [ n v n y a n y N Direct Form I implementation
On the -domain or equivalently M N b a H H H 0 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 X b X H V M ) ( ) ( ) ( ) ( ) ( 2 V a V H Y N
By changing the order of H and H 2, onsider the equivalence on the -domain: where Let Then ) ( ) ( ) ( 2 H H H M b H 0 ) ( ) ( ) ( 2 N a H ) ( ) ( ) ( ) ( ) ( 2 X a X H W N ) ( ) ( ) ( ) ( 0 W b W H Y M
In the time domain, We have the following equivalence for implementation: N n x n w a n w ] [ ] [ ] [ M n w b n y 0 ] [ ] [ We assume M=N here
Note that the exactly the same signal, w[], is stored in the two chains of delay elements in the bloc diagram. The implementation can be further simplified as follows: Direct Form II (or Canonic Direct Form) implementation
By using the direct form II implementation, the number of delay elements is reduced from (M+N) to max(m,n). Example: H ( ).5 2 0.9 2 Direct form I implementation Direct form II implementation
Representing by signal-flow graph Example: the signal-flow graph of direct form II.
Example: H 2 2 ( )( ) ( ) 2 0.75 0.25 ( 0.5 )( 0.25 ) Cascade structure: direct form I implementation Cascade structure: direct form II implementation
Cascade Form From the fundamental theorem of algebra, we now that an n-th order real-coefficient polynomial has n roots that are real or complex-conjugate pairs. From this theorem, it ensures that we can factor the numerator and denominator polynomials. We can express H() in the form: H M M 2 ( f ) ( ) N N 2 ( c ) ( ( g d )( )( g d ) ) where M=M +2M 2 and N=N +2N 2, g and g * are a complex conjugate pair of eros, and d and d * are a complex conjugate pair of poles.
A general form is where and we assume that MN. The real poles and eros have been combined in pairs. If there are an odd number of eros, one of the coefficients b 2 will be ero. s N a a b b b H 2 2 2 2 0 ) ( 2 / ( ) N N s
It suggests that a difference equation can be implemented via the following structure consisting of a cascade of second-order and first-order systems: cascade form of implementation (with a direct form II realiation of each second-order subsystems)
Parallel Form We also now that a rational function can be represented as sum of partial fractions. If we represent H() by additions of low-order rational systems: where N=N +2N 2. If MN, then N p = MN. 2 0 ) )( ( ) ( ) ( N N N d d e B c A C H p
Parallel Form Alternatively, we may group the real poles in pairs, so that p N s N a a e e C H 2 2 0 0 ) (
Illustration of parallel-form structure for six-order system (M=N=6) with the real and complex poles grouped in pairs.
Example: consider still the same system 2 2 2 0.25 0.75 8 7 8 0.25 0.75 2 ) ( H
another alternation of the same system H ( ) 0.75 2 2 0.25 2 8 8 0.5 25 0.25 Hence, given a system function, there are many ways to implement it. There are equivalent when infinite-precision arithmetic is used. However, their behavior with finiteprecision arithmetic could be quite different.
Remar: While the signal flow graph is an efficient way to represent a difference equation, not all of its instances are realiable: If a system function has poles, a corresponding signal flow graph will have feedbac loops. A signal flow graph is computable if all loops contain at least one unit delay element. Eg. A non-computable system Computable systems
Discrete Fourier Transform (DFT) Currently, we have investigated three cases of Fourier transform, Fourier series (for continuous periodic signal) Continuous Fourier transform (for continuous signal) Discrete-time Fourier transform (for discrete-time signal) All of them have infinite integral or summation in either time or frequency domains.
There is still another type of Fourier transform Consider a discrete sequence that is periodic in the time domain. Eg., it can be obtained by a periodic expansion of a finite-duration sequence, ie., we image that a finitelength sequence repeats, over and over again, in the time domain). Then, in the frequency domain, the spectrum shall be both periodic and discrete, ie, the frequency sequence is also made up of a finite-length sequence, which repeats over and over again in the frequency domain.
Discrete Fourier Transform Considering both the finite-length (or finite-duration) sequences in one period of the time and frequency domains, leads to a transform called discrete Fourier transform (DFT). In principle, DFT acts lie performing multiple (a ban of) FIR filters simultaneously at the same time.
DFT Definition If x[n] is a finite-length sequence (n0 only when n <N), its DTFT X(e jw ) is still a periodic continuous function with period 2. The DFT of x[n], denoted by X() is as follows: j 2 / N where, and W n are the roots of W N =. W e
Relation to DTFT for Finite-length sequence X() is the uniform samples of the DTFT X(e jw ) at the discrete frequency w = (2/N), when the frequency range [0, 2] is divided into N equally spaced points. Note that the above explains only the relation between DFT and DTFT for the simple case that x[n] is a finite-length sequence. We will investigate more in-depth relationships between DFT continuous FT (and also between DFT and DTFT) for length-unlimited sequences in the future.
From Kuhn 2005
From Kuhn 2005
Four types of Fourier Transforms Frequency domain nonperiodic Frequency domain periodic Time domain nonperiodic Continuous Fourier transform (both domains are continuous) DTFT (time domain discrete, frequency domain continuous and band-limited or periodic) Time domain periodic Fourier series (time domain continuous and periodic, frequency domain discrete) DFT/DTFS (time domain discrete, frequency domain discrete, and both finite-duration or periodic)