香港科大信号系统英文课件Key Points in Signals and Systems-Chapters 7,9

Key Points in Signals and Systems4. Sampling Theorem4.1 Sampling Theorem: A band-limited signal x (t ) with bandwidth (unilateral) M ωrad/sec can be uniquely recovered from its samples x (nT ) if the sampling interval T is sufficiently small: M T ωπ/<. It is easier to think in terms of ordinary frequencies: the sampling frequency must be greater than two times the unilateral bandwidth. That is:πωM M s f T f =>=21. 4.2 Spectrum of sampled impulse train is a Poisson sum of original spectrum:))2((1)()()()()()(∑∑∞-∞=∞-∞=-=⇒-==k p n p T k j X T j X nT t nT x t p t x t x πωωδ If T is sufficiently small, there is no aliasing and Tj X T j X p πωωω<=||for )(1)(, so low-pass filtering of )(ωj X p (and scale by T ) reproduces )(ωj X .4.3 Generalizing 4.2, multiplying x (t ) with any T -periodic signal q (t ) gives:))2(()()()()(∑∞-∞=-=⇒=k k q q T k j X c j X t q t x t x πωωwhere k c are the FS coefficients of q (t ) So low-pass filtering )(t x q reproduces c 0x (t ) if period T is small enough to avoid aliasing.4.4 Spectrum of DT sampled signal is a frequency-scaled version of )(ωj X p)()()()(][Ω=Ω=⇒=Ωs p p j d d jf X TjX e X nT x n x Frequency range of a DT signal is ππ≤Ω<-, but the actual frequency represented is scaled by the (ordinary) sampling frequency: Ω=s f ω. So if the sampling frequency is 1 MHz, the actual frequency represented in the DT signal is from -0.5 MHz to 0.5 MHz.4.4 Discrete Approximation of Signals and SystemsIf signal x (t ) and system h (t ) are band-limited, )(t Tx p produces same output as x (t ) because )()()()2((1)()(ωωωπωωωj H j X j H T k j X T T j H j TX k p =⎪⎭⎫ ⎝⎛-=∑∞-∞= Likewise, if signal to be processed and system are band-limited, we can approximate impulse response h (t ) by a discrete version ∑∞-∞=-=n p nT t p t Th t Th )()()(. The SAW filter (serviceacoustic wave filter) in our cell phone is based on this principle.4.5 Understanding Aliasing in real-life: When sampled at T f s /1= Hz, complex sinusoids that rotate at f Hz and at f mf s +Hz are identical: nT f mf j fnT j s e e )(22+≡ππ because ππ22mn nT mf s =. So in a movie shown at 30 frames/sec, a wagon wheel turning at 31 Hz will appear as turning at 1 Hz. A wheel turning at 29 Hz will appear as turning backward at 1 Hz.When sampled at T f s /1= Hz, real sinusoids that rotate at f Hz and at f mf s ±Hz are identical: nT f mf fnT s )(2cos 2cos ±≡ππbecause )cos(cos θθ-=. Sampling rate in the telephony networkis 8,000 Hz. Therefore, inputs at 7,000Hz, 9,000 Hz and 15,000 Hz will all appear as 1,000 Hz.Moire effect in images. Examples of Anti-Aliasing Filters.5. Differential Equations as LTI Systems5.1 Frequency response of ODE as LTI is of rational form5.2 Determine impulse response by factorizing )(ωj D followed by partial fraction expansion toexpress )(ωj H in partial fraction form )()()(11t u e t h j c j H n k n k k k k ∑∑===⇒-=ααωω.5.3 Roots of )(ωj D , the s k 'α, characterize the system. Any k αhas positive real part, system is unstable. Any k αhas imaginary part, system is oscillatory. Also, for real system,k α’s and k c ’s must come in conjugate pairs.5.4 2nd order system is stable only if all coefficients of denominator are all the same sign. Also, we can alternatively express the denominator polynomial of a 2nd order system in terms of itsnatural frequency n ωand parameter ς:22012)(2)()()()(n n j j a j a j j D ωωςωωωωω++=++=.1≅ς system critically damped;1>>ς system over-damped;+→0ςsystem under-damped;10<<ς system oscillatory;0<ς system unstable.∑∑∑∑=-=-====⇒=N k k k N k k k k k k N k k k k N k j a j b j D j N j H dt t x d b dt t y d a 010100)()()()()()()(ωωωωω6. Laplace Transform6.1 Two ways to look at relationship between FT and LT:-ωωj s s X j X ==)()(: FT is one cross-section of LT. FT exists only if signal/system is stable-})({)(t e t x FT j s X σωσ-=+=: LTat ωσj s += is FT of signal first multiplied by t e σ-6.2 ROC of LT in rational form is either the whole s -plane, a RHP, a LHP, or a vertical strip on the s -plane. ROC is bounded by poles.6.3 For LT in rational form, we take right-sided or left-sided inverse transform for each partial fraction term depending on whether the ROC is to the right or to the left of the corresponding pole:For each term, take right sided inverse FT if ROC is to the right of k αand vice versa6.4 Signal/system is stable if ROC contains ωj -axis (FT exists). Signal/system is causal iff rational LT has ROC that is a RHP. Stable causal system must have all poles left of ωj -axis. 6.5 Differentiation property )()(s sX t x dt d L −→←. We can re-express LTI ODE in terms of Laplace Transform 6.6 Geometric Evaluation and Filter design by pole placement)(k j αω-and )(i j βω-can all be viewed as 2-D vectors on the s -plane. To make |)(|ωj H large around 1ω, we place pole(s) near 1ωj so ||k j αω- in the denominator becomes small. Real-part of pole determines half-power width.6.7 Butterworth FilterLower-pass filter: distribute N poles evenly along left semi-circle with radius c ωcentered at the origin. Large N for fast roll-off from pass-band to stop-band.Band-pass filter: distribute N Poles evenly along left semi-circle with radius c ωcentered at the intended pass-band center frequency on ωj -axis.6.8 Block Diagram Implementation of rational/differential equation systems- Simple feedback circuit for 1st order prototypical system- Numerator polynomial as purely feed-forward system- Higher order system in direct, product, and parallel forms. ∑=-----=+++++++=n k k k n n n n n s c a s a s a s b s b s b s X 101110111........)(α∑∑∑∑=-=-====⇒=N k k k N k k k k k k N k k k k N k s a s b s D s N s H dt t x d b dt t y d a 010100)()()()()(∏∏∏∏=-==-=------=⇒--=+++++++=n k k n i i n k k n i i n n n n n j j M j H s s M a s a s a s b s b s b s H 11111101110111|||||)(|)()(........)(αωβωωαβ。