Some key points Any arbitrary input sequence xn can be expressed as a linear combination of delaved and advanced unit sample sequences xn]=∑xk][n-k] 2..Linear Time-Invariant (LTD System A system satisfying both the linearity and the time-invariance property. If yiIn] is the output due to an input xin and yiN is the output due to an input x2lnl then for an input xn]=axiN+bx2InI the output is given b yIn]=ayiIn+by2InI Hence, the above system is linear Above property must hold for any arbitrary constants a and b and for all possible inputs xlIn and xiN. For a shift-invariant system, if yiIn is the response to an input x1Inl, then the response to an input xIn]=xiIn-nol is simply yIn]=yiln-nol where no is any positive or negative integer The above relation must hold for any arbitrary input and its corresponding output The above property is called time-invariance property, or shift-invariant proterty For example, Down-sampling x,n=xnM is time-variant 3. Convolution in time domain Ifa signalxIn] inputs to a system with the impulse response h[n], the output signal 4*小=∑m师n-m The main idea of the method is that an input signal is represented as( decomposed) the sum
Some key points 1. Any arbitrary input sequence x[n] can be expressed as a linear combination of delayed and advanced unit sample sequences 2. •Linear Time-Invariant (LTI) System A system satisfying both the linearity and the time-invariance property. •If y1[n] is the output due to an input x1[n] and y2[n] is the output due to an input x2[n] then for an input x[n]=ax1[n]+bx2[n] the output is given by y[n]=ay1[n]+by2[n] •Hence, the above system is linear . •Above property must hold for any arbitrary constants a and b and for all possible inputs x1[n] and x2[n]. •For a shift-invariant system, if y1[n] is the response to an input x1[n] , then the response to an input x[n]=x1[n-n0] is simply y[n]=y1[n-n0] where n0 is any positive or negative integer •The above relation must hold for any arbitrary input and its corresponding output •The above property is called time-invariance property, or shift-invariant proterty - For example, Down-sampling x [n] x[nM ] d = is time-variant 3．Convolution in time domain If a signal x[n] inputs to a system with the impulse response h[n] ,the output signal =− = = − m y[n] x[n] h[n] x[m]h[n m] The main idea of the method is that an input signal is represented as(decomposed) the sum = − k =− x[n] x[k][n k]
of basic signal: S[n], the response of LTI system is the synthesis of basic response hn 4. Discrete-Time Signals In the transform domain (1)From Ft TO DTFT and DFT ↑x(t) FI ↑x[nT XGo) DTFT q(t =2x/T :, Time domain Frequency domain Continue aperiodical fFT- Continue aperiodical Periodical ←FST→ discrete spectrum ←DTH→ periodical spectrun Discrete periodical +DFT+ periodical discrete The DTFT X(e w)of a sequence x n is a continuous function of o lt is also a periodic function of o with a period 2. DTFT is the Fourier Transform of discrete-time sequence. It is discrete in time domain and its spectrum is periodical (2)Relationships ZT and DTFT and DFT A finite-length sequence xin0≤n≤N-1, x[n]→X(=) xIn I
of basic signal: [n],the response of LTI system is the synthesis of basic response h[n] . 4. Discrete-Time Signals In the Transform Domain (1) From FT TO DTFT and DFT Time domain Frequency domain Continue aperiodical FT → Continue aperiodical Periodical FST → discrete spectrum Discrete DTFT → periodical spectrum Discrete periodical DFT → periodical discrete •The DTFT X(ejw) of a sequence x[n] is a continuous function of •It is also a periodic function of with a period 2DTFT is the Fourier Transform of discrete-time sequence. It is discrete in time domain and its spectrum is periodical. (2) Relationships ZT and DTFT and DFT A finite-length sequence x[n];0 n N −1, x[n] X (z) = − = − 1 0 ( ) [ ] N n n X z x n z . x(t) X(jω) P(jω) ω0 ω0=2π/Ts P(jω) ω0 ω0=2π/Ts x[nT] X(jω) q(t) T FT DFT DTFT Q(jω) Ω0 … … Ω0= 2π/T Q(jω) Ω0 … … Ω0= 2π/T Q(jω) Ω0 … … Ω0= 2π/T Q(jω) Ω0 … … Ω0= 2π/T Q(jω) Ω0 … … Ω0= 2π/T P(t) Ts … …
Xe)=∑xln/e -k when 2==k=e X(x)=∑xn=∑xnpw如= DFTIxn] That means: The ZT on the unit circle in Z-plane is the dtFtof x[n. The samples on the unit circle in Z-plane. X(E,), are the det of xn 5. The Concept of Filtering One application of an lti discrete-time system is to pass certain frequency components in an input sequence without any distortion(if possible)and to block other frequency components In another words, the sinusoidal components of the input, some of these components can be selectively heavily attenuated or filtered with respect to the others For example, a signal is inputted to a lowpass filter. If we change the frequency response of the filter, the output signal will be changed (as shown in following figure) A Lowpass 6. Analog Lowpass Filter Specifications passbandedge frequency: P stopband edge frequency: S Peak pa pp an=-20l0g0(-)B Minimum stopband attenuation: as=-20log1o(os)laB
= =− − n jω jnω X(e ) x[n]e when k N k N j z z k e W − = = = 2 , = = − = 1 − 0 2 ( ) [ ] N n kn N j k X z x n e [ ] [ [ ]] 1 0 x n W DFT x n N n kn N = − = That means: The ZT on the unit circle in Z-plane is the DTFT of x[n] . The samples on the unit circle in Z-plane, ( ) k X z ,are the DFT of x[n]. 5. The Concept of Filtering One application of an LTI discrete-time system is to pass certain frequency components in an input sequence without any distortion (if possible) and to block other frequency components。In another words,the sinusoidal components of the input, some of these components can be selectively heavily attenuated or filtered with respect to the others。 For example,a signal is inputted to a lowpass filter.If we change the frequency response of the filter,the output signal will be changed(as shown in following figure). A Lowpass Filter 6. Analog Lowpass Filter Specifications •passband edge frequency： P •stopband edge frequency： S •Peak passband ripple ： 20log (1 )[ ] p = − 10 − p dB •Minimum stopband attenuation： 20log ( )[ ] S = − 10 s dB
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