Interchange of Filtering and Downsampling/Upsampling Downsampling and upsampling are linear systems, but not LTI systems. They cannot be implemented by difference equations, and so we cannot apply z-transform for their representation. However, they have apparent properties (frequency expansion and concentration ), and so frequency-domain analysis can still be performed. In particular, we can show that the following two systems are equivalent equivalent systems
equivalent systems Why? In the frequency domain Also So
Hence A similar identity applies to upsampling equivalent systems
Polyphase Decomposition Polyphase decomposition of a sequence is obtained by representing it as a superposition of M sequences (0 k<m) Note that h[-m] 0 0 0 h[0] 0 0 0 h[m] 0 0 0 h[2m].... 0 h[-m+1] 0 0 0 h[1] 0 0 0 h[m+1] 0 0 0 +) h(-m) h(-m+1) h[0] h[1] h[m] h[m+1].h[2m]
Define e k [n] to be which are referred to as the polyphase components of h[n]. By z-transform and downsampling, we can reconstruct h[n] by the following figure
Since the delays (z -i ) can be chained, we can also reconstruct h[n] by the following equivalent figure:
Consider the red-square part:
From this part, we have the relationship of the inputs e k [n] (k= 0 to M-1) and the output h[n]. In the Z-domain, the relationship becomes: It means that, the system (filter) H(z) can be decomposed into M parallel filters (addition of these filters). More specifically, the system function H(z) is expressed as a sum of delayed polyphase component filters.
Hence, when input x[n] is applied to the system H(z), its diagram structure can be represented as The time-domain interpretation of the above is straightforward. If H(z) is an FIR filter, then for each component, the required clock rate ( 時脈 ) is M-times slower than that of the input. Less cost in digital circuits design.
Polyphase Implementation of Decimation Filters Remember that, when performing downsampling, it is usually required a low-pass filter H(z) being performed beforehand, and the entire process is called decimation. (A) By decomposing the system function H(z) by its polyphase components, and note that the downsampling can be combined with each component, we have the diagram as follows:
Implementing the decimation filter using polyphase decomposition
The above system performs downsamplings after the filtering, which is inefficient. According the interchange property of downsampling and filter, we have the following equivalent system that performs downsamplings beforehand: (B)
Advantage: If H(z) is an N-point FIR filter In the original design (Figure A) for the decimation filter, it needs N multiplications and N-1 additions per unit time. In the polyphase-decomposition design (Figure B) for the decimation filter, each filter E k (z) is of length N/M. Each filter requires, per unit time, (1/M)(N/M) multiplications and (1/M)((N/M) - 1) additions. Hence, Figure B requires N/M multiplications and (N/M)-1+M-1 additions. per unit time. Polyphase decomposition principle has been used for music recording (such as polyphase suband coding in MP3), and DSP ASIC(Application Specific IC) for CD players, such as NPC-SM5813.
Polyphase Implementation of Interpolation Filters Similar to the decimation case, polyphase decomposition principle can be applied to the interpolation filter too. NL multiplications and NL-1 additions per unit time L(N/L) multiplications and L((N/L)-1) + L-1 additions per unit time
Correlation Given a pair of sequences x[n] and y[n], their cross correlation sequence is r xy [l] is defined as r xy [ ] [ ] [ ] l = n= x y n l for all integer l. The cross correlation sequence can help measure similarities between two signals. For power signal: n
Cross correlation is very similar to convolution, unless the indices changes from l n to n l. Relation between cross correlation and convolution: Autocorrelation For power signal: r xx [ l] = x[ n] x[ n l] n=
n= 2 = Properties Consider the following non-negative expression: a ( [ ] [ ]) 2 2 2[ ] 2 [ ] [ ] 2 ax n y n l = a x n + a x n y n l + y [ n l] r xx That is, + [ 0] + 2ar [ l] + r [ 0] 0 xy [ a ] Thus, the matrix Its determinate is nonnegative. yy n= [ 0] r [ ] xy l [ l] r [ 0] rxx a 1 0 r xy yy 1 r r xx xy [ 0] r [ ] xy l [ ] [ ] l ryy 0 n= for all a n= is positive semidefinite.
The determinant is r xx [0]r yy [0] r xy2 [l] 0. Properties r xx [0]r yy [0] r xy2 [l] r xx2 [0] r xx2 [l] Normalized cross correlation and autocorrelation: ρ xx [ ] xx [ l] [ ] rxx l = ρ = r The properties imply that ρ xx [0] 1 and ρ yy [0] 1. This property can also be explained by Schwartz inequality. Consider x and y to be two infinite-length vectors. Then r xx [0] and r yy [0] are the squared lengths of x and y, respectively. r xy [l] is the inner product between x[n] and y[n-l]. From Schwartz inequality, similar property holds for the inner product of x and any vector consisting of a reshuffle of the elements of y. [ l] [ ] xy xy l 0 rxx yy r [ 0] r [ 0]
For two signals having the same autocorrelations, what s their common property? Property in the frequency domain The DTFT of the autocorrelation signal r xx [l] is the squared magnitude of the DTFT of x[n], i.e., X(e jw ) 2. That is Hence, signals with the same autocorrelation share the same magnitude spectrum in the frequency domain, albeit their phases are different. An example of describing the property of a class of signals. Autocorrelation is quite often be used for finding the period of a periodical signal. By definition, autocorrelation peaks at the integer multiples of the period. Correlation is useful in random signal modeling and processing
DTFT continue (c.f. Shenoi, 2006) We will represent the spectrum of DTFT either by H(e jwt ) or more often by H(e jw ) for convenience. When represented as H(e jw ), it has the frequency range [- π,π]. In this case, the frequency variable is to be understood as the normalized frequency. The range [0, π] corresponds to [0, w s /2] (where w s T=2π), and the normalized frequency π corresponds to the Nyquist frequency (and 2π corresponds to the sampling frequency).
DC response When w=0, the complex exponential e jw becomes a constant signal, and the frequency response X(e jw ) is often called the DC response when w=0. The term DC stands for direct current, which is a constant current. DTFT Properties Revisited Time shifting
Frequency shifting Time reversal
DTFT of δ(n) n δ n= ( ) jwn n e = e jw0 =1 DTFT of δ(n+k)+ δ(n-k) According to the time-shifting property, DTFT of δ ( ) jwk ( ) jwk n + k is e, DTFT of δ n k is e Hence ( ) ( ) jwk jwk n + k + δ n k is e + e 2cos( wk) DTFT of δ =
DTFT of x(n) = 1 (for all n) x(n) can be represented as We prove that its DTFT is
Hence = π π = π π π π δ k = k = ( ) jwn w k e dw δ 2π ( w k) dw δ 2π ( w) dw = δ ( w) dw + δ ( w) dw + δ ( w)dw π π π π = δ ( w ) dw =1 for all n
From another point of view According to the sampling property: the DTFT of a continuous signal x a (t) sampled with period T is obtained by a periodic duplication of the continuous Fourier transform X a (jw) with a period 2π/T = w s and scaled by T. Since the continuous F.T. of x(t)=1 (for all t) is δ(t), the DTFT of x(n)=1 shall be a impulse train (or impulse comb), and it turns out to be DTFT of a n u(n) let ( a <1) then This infinite sequence converges to when a <1.
DTFT of Unit Step Sequence
Adding these two results, we have the final result
Differentiation Property
DTFT of a rectangular pulse
and get
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 still is 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 finite-length 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 finitelength sequence, which repeats over and over again in the frequency domain. 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.
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) Time domain periodic Fourier series (time domain continuous, frequency domain discrete) DFT/DTFS (time domain discrete, frequency domain discrete, and both finite-duration)
DFT and DTFT A closer look We discuss the DTFT-IDTFT pair ( I means inverse) for a discrete-time function given by and The pair and their properties and applications are elegant, but from practical point of view, we see some limitations; eg. the input signal is usually aperiodic and may be infinite in length.
Example of a finite-length x(n) and its DTFT X(e jw ). A finite-length signal Its magnitude spectrum Its phase spectrum
The function X(e jw ) is continuous in w, and the integration is not suitable for computation by a digital computer. We can discretize the frequency variable and find discrete values for X(e jw k ), where wk are equally sampled whthin [-π, π]. Discrete-time Fourier Series (DFS) Let x(n) (n Z) be a finite-length sequence, with the length being N; i.e., x(n) = 0 for n < 0 and n > N. Consider a periodic expansion of x(n): x p ( n + KN ) = x( n), n = 0,1,..., n 1, K is any integer x p (n) is periodic, so it can be represented as a Fourier series:
To find the coefficients X p (k) (with respect to a discrete periodic signal), we use the following summation, instead of integration: First, multiply both sides by e -jmw 0k (w 0 =2π/k), and sum over n from n=0 to n=n-1: By interchanging the order of summation, we get
Noting that pf. When n=m, the summation reduces to N When n m, by applying the geometric-sequence formula, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 / 2 ) ( / 2 ) ( / 2 2 / 2 ) ( / 2 ) ( / 2 1 ' ' / 2 1 0 / 2 = = = = + = = k N j m k N j m k N j k j k N j m k N j m N k N j m N m n kn N j N n m n k N j e e e e e e e e π π π π π π π π π 1, 1 1 = + = r r r r r M N N M n n
Since there is only one nonzero term, = X p (k)n The final result is The following pairs then form the DFS
Relation between DTFT and DFS for finite-length sequences We note that DTFT spectrum DFS coeficient Main property: In other words, when the DTFT of the finite length sequence x(n) is evaluated at the discrete frequency w k = (2π/N)k, (which is the kth sample when the frequency range [0, 2π] is divided into N equally spaced points) and dividing by N, we get the Fourier series coefficients X p (k).
A discretetime periodic signal Its magnitude spectrum (sampled) Its phase spectrum (sampled)
To simplify the notation, let us denote W N = e ( 2π / N ) The DFS-IDFS ( I means inverse ) can be rewritten as (W=W N ) j Discrete Fourier Transform (DFT) Consider both the signal and the spectrum only within one period (N-point signals both in time and frequency domains) IDFT (inverse DFT) DFT
Relation between DFT and DTFT: The frequency coefficients of DFT are the N-point uniform samples of DTFT with [0, 2π]. The two equations DFT and IDFT give us a numerical algorithm to obtain the frequency response at least at the N discrete frequencies. By choosing a large value N, we get a fairly good idea of the frequency response for x(n), which is a function of the continuous variable w. Question: Can we reconstruct the DTFT spectrum (continuous in w) from the DFT? Since the N-length signal can be exactly recovered from both the DFT coefficients and the DTFT spectrum, we expect that the DTFT spectrum (that is within [0, 2π]) can be exactly reconstructed by the DFT coefficients.
Reconstruct DTFT from DFT (when the sequence is finite-length) By substituting the inverse DFT into the x(n), we have a geometric sequence
By applying the geometric-sequence formula
So The reconstruction formula
In summary If x[n] is a finite-length sequence (n 0 only when n <N), its DTFT X(e jw ) shall be a periodic continuous function with period 2π. The DFT of x[n], denoted by X(k), is also of length N. W = e j( 2π / N )n where, and W n are the the roots of W n = 1. Relationship: X(k) is the uniform samples of X(e jw ) at the discrete frequency w k = (2π/N)k, when the frequency range [0, 2π] is divided into N equally spaced points.