Sequences Dirac’s Delta Function ( 1 0 x 6= 0 fxngn=−∞ is a discrete sequence. δ(x) = , R 1 δ(x)dx = 1 1 x = 0 −∞ Can derive a sequence from a function x(t) by having xn = n R 1 x(tsn) = x( ). Sampling: f(x)δ(x − a)dx = f(a) fs −∞ P1 P1 Absolute summability: n=−∞ jxnj < 1 Dirac comb: s(t) = ts n=−∞ δ(t − tsn), x^(t) = x(t)s(t) P1 2 Square summability: n=−∞ jxnj < 1 (aka energy signal) Periodic: 9k > 0 : 8n 2 Z : xn = xn+k Fourier Transform ( 0 n < 0 j!n un = is the unit-step sequence Discrete Sejong’s of the form xn = e are eigensequences with 1 n ≥ 0 respect to a system T because for any !, there is some H(!) ( such that T fxng = H(!)fxng 1 n = 0 1 δn = is the impulse sequence Ffg(t)g(!) = G(!) = R g(t)e∓j!tdt 0 n 6= 0 −∞ −1 R 1 ±j!t F fG(!)g(t) = g(t) = −∞ G(!)e dt Systems Linearity: ax(t) + by(t) aX(f) + bY (f) 1 f Time scaling: x(at) jaj X( a ) 1 t A discrete system T transforms a sequence: fyng = T fxng. Frequency scaling: jaj x( a ) X(af) −2πjf∆t Causal systems cannot look into the future: yn = Time shifting: x(t − ∆t) X(f)e f(xn; xn−1; xn−2;:::) 2πj∆ft Frequency shifting: x(t)e X(f − ∆f) Memoryless systems depend only on the current input: yn = R 1 2 R 1 2 Parseval’s theorem: 1 jx(t)j dt = −∞ jX(f)j df f(xn) Continuous convolution: Ff(f ∗ g)(t)g = Fff(t)gFfg(t)g Delay systems shift sequences in time: yn = xn−d Discrete convolution: fxng ∗ fyng = fzng () A system T is time invariant if for all d: fyng = T fxng () X(ej!)Y (ej!) = Z(ej!) fyn−dg = T fxn−dg 0 A system T is linear if for any sequences: T faxn + bxng = 0 Sample Transforms aT fxng + bT fxng Causal linear time-invariant systems are of the form The Fourier transform preserves function even/oddness but flips PN PM real/imaginaryness iff x(t) is odd. k=0 akyn−k = m=0 bmxn−m, but all of them can be rep- resented with only one set of coefficients, i.e. bi = δi. In this 1 1 Ffcos(2πf0t)g(f) = 2 δ(f − f0) + 2 δ(f + f0) case the sequence fang is called the impulse response of T since Ffsin(2πf t)g(f) = − j δ(f − f ) + j δ(f + f ) fang = T fδng. 0 2 0 2 0 P1 P1 n Ffts δ(t − tsn)g(f) = δ(f − ) n=−∞ n=−∞ ts Convolution Aliasing P1 fpng ∗ fqng = frng () 8n 2 Z : rn = k=−∞ pkqn−k Due to overlapping in the frequency domain, the Nyquist limit stipulates that frequencies in the signal should be limited so Associativity: (fpng ∗ fqng) ∗ frng = fpng ∗ (fqng ∗ frng) fs that jfj ≤ 2 . Bandpass sampled signals can also be re- Commutativity: fpng ∗ fqng = fqng ∗ fpng constructed if their spectral component lie within the interval fs fs n 2 < jfj < (n + 1) 2 Linearity: fpng ∗ faqn + brng = a(fpng ∗ fqng) + b(fpng ∗ frng) An ideal filter for enforcing the Nyquist criterion is rect(tsf) = Identity: fp g ∗ fδ g = fp g ( f n n n 1 jfj < s 2 −1 sin πtfs withF frect(f)g(t) = fs . Shifting: fp g = fp g ∗ fδ g fs πtfs n−d n n−d 0 jfj > 2 Sine-wave and exponential sequences form a family of discrete Typically the sampling frequency is chosen so there is a transi- sequences that is closed under convolution with arbitrary se- tion band between the Nyquist limit and the highest frequency quences. expected to be sampled. 1 i Discrete Fourier Transform Hamming: wi = 0:54 − 0:46 cos(2π n−1 ) Since a windowed signal has been forced to 0 outside the sam- n−1 ik P −2πj n Xk = i=0 xie pled region, it may be zero-padded at either end without further 1 n−1 2πj ik distortion. However, that does increase the frequency resolution x = P X e n k n i=0 i of the DFT (albeit without adding any new information!). f The elements X to X n contain the frequency components 0, s , 0 2 n 2fs , :::, fs n 2 Filter Design We can make use of the fast Fourier transform: ik n −1 2ik n−1 Pn−1 −2πj P 2 −2πj Fnfxig = xie n , so = x2ie n + We can turn a spectrum X(f) into an inverted spectrum i=0 i=0 i=0 f f k n 2ik 0 s s −2πj 2 −1 −2πj X (f) = X( − f) we shift the spectrum by . This can e n P x e n . This repeated subdivision has 2 2 i=0 2i+1 be done by multiplying with the sequence with y = cos(πi). log n rounds and n log n additions and multiplications overall, i 2 2 This can turn a low-pass filter into a high-pass filter. compared to n2 for the equivalent matrix multiplication. The ideal low-pass filter has two problems in that it is not causal By the symmetry properties of the FFT, 8i : x = R(x ) () i i and has an infinitely long impulse response. This is solved by 8i : X = X? and 8i : x = jI(x ) () 8i : X = −X?. n−i i i i n−i i applying a windowing function and a delay to make it causal, We can therefore compute the FFT of two real valued sequences such as: by computing that of x0 + jx00 and saying X0 = 1 (X + X? ) i i i 2 i n−1 f 0 1 ? sin(2π(i− n ) c ) and X = (X − X ). fc 2 fs i 2 i n−1 hi = 2 n f wi for an nth order low pass filter. fs 2π(i− ) c 2 fs We can also compute complex multiplication with 3 multipli- cations and 5 additions rather than 4 multiplications and 2 additions that are required naively if we have the numbers in Z-Transform Cartesian format. P1 −n If X(z) = n=−∞ xnz , which defines a complex-valued sur- face over . For finite sequences, this surface is defined every- C Application to Convolution xn+1 where, else it converges in the region limn!1 < jzj < xn xn+1 We can compute convolution using FFT with O(m log m + limn→−∞ . xn n log n) rather than mn multiplications if we multiply in the Pk Pm Fourier domain instead. Note however that the sequences be- For an LTI system defined by l=0 alyn−l = l=0 blxn−l, −1 −2 −m b0+b1z +b2z +:::+bmz ing convolved must be periodic with equal period lengths. If H(z) = −1 −2 −k so that fyng = fhng ∗ fxng. a0+a1z +a2z +:::+akz this condition is not fulfilled the sequences must be zero-padded This has m zeroes and k poles at non-zero locations in the z to a length of at least m + n − 1 to ensure the start and end of plane, plus k − m zeroes (if k > m) or m − k poles (if m > k) the resulting sequence do not overlap. at z = 0. We can use this result to perform deconvolution by dividing the The order of a filter is the number of zeroes it has. Possible −1 Ffsg filter types are Butterworth, Chebyshev Type I/Type II and Fourier representation: u = F f Ffhg g Elliptic. Spectral Estimation Random Sequences If the input to the DFT is sampled but not periodic, the DFT may still be used to calculate an approximation. However, we In such a sequence every value is the outcome of a one random see leakage of energy to frequency bins that are not strictly variable from a corresponding sequence. This collection of ran- present in the input signal but adjacent to expressed frequen- dom variables is called a random process. Such a process is sta- cies. The peak amplitude also changes as the frequency of a tionary if P (a ; : : : ; a ) = P (a ; : : : ; a ) xn1+l;:::;xnk+l 1 k xn1;:::;xnk 1 k tone changes from one bin to the next, reaching its lowest am- where P (a) = P rob(x ≤ a). plitude of 62% halfway between the two. This is known as xn n R scalloping. It occurs since the effective window introduced by Expected value: mx = "(xn) = apxn (a)da the DFT is rectangular, which corresponds to convolution with 2 2 2 Variance: V ar(xn) = "(jxn − "(xn)j ) = "(jxnj ) − j"(xn)j sinc in the frequency domain. ? Correlation: Cor(xn; xm) = "(xnxm) = φxx(0) A non-periodic signal can have a windowing function applied ? to try and reduce scalloping and leakage. Possibilities include: Cross-correlation: φxy(k) = "(xn+kyn), Autocorrelation: φxx(k) 2i Triangular: wi = 1 − 1 − n ? Cross-covariance: γxy(k) = "[(xn+k − mx)(yn − my) ] = i ? Hanning: wi = 0:5 − 0:5 cos(2π n−1 ) φxy(k) − mxmy, Autocovariance: γxx(k) 2 P1 Deterministic cross-correlation: cxy(k) = i=−∞ xi+kyi, so Recall that ~x is an eigenvector if for some λ 2 R A~x = λx. Now T that fcxy(k)g = fxkg ∗ fy−kg. This implies that the Fourier any symmetric matrix A can be represented as UΛU where transform Cxx(f) is identical to the power spectrum. Λ = diag(λ1; : : : ; λn) of increasing λ and the columns of U are P1 the corresponding orthonormal eigenvectors.