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Although all CT sinusoidals are periodic, their DT counterparts x[k] =A sin(wk + ) may not always be periodic. In other words, x[k] is periodic if we can find a set of values for m, K0 Z+. DT sinusoidal sequence x[k] = A sin(_0k + ) is periodic iff _0/2 is a rational number. if you sample a CT periodic signal, the DT signal need not always be periodic. The signal will be periodic only if you choose a sampling interval T such that the term 0T/2 is a rational number. periodic non-sinusoidal signal can be expressed as a linear combination of a sine wave having the same fundamental frequency as the fundamental frequency of the original periodic signal and the harmonics of the sine wave. Power signal has infinite energy and energy signal has zero power. Most periodic signals are power signals. Periodic DT sinusoid signal of the form x = a cos (wk + theta) has avg power = a^2 /2. Rectgular pulse rect (t/tau) = 1 if |t| tau/2. For DT, rect (k/ 2N+1) = 1 if |k|N. CT EXPONENTIAL FUNCTION..real comp=0, sigma =0
Cos x = (1+cos2x) Complex valued exp is periodic iff w0/ 2pie is rational. Causal signals signals starting at time t=0. Causal exponential function e^(st) * u(t). impulse function is even, multiplication of arbitrary func with impulse results in impulse func with area equal to value of arbitrary func at the location of impulse. Interpolation: inserts m-1 zeros in between adjacent samples of the
Sequence. It is reversible, can get x[k] from xmk. Chapter 2 linear CT system repped by linear constant coefficient diff. eq of the second order. R in series with l and c. output across c. r connected to v(t). ir = (y-v)/R. il = 1/L (integrate y(tau) from negative infinity to t). ic = C . dy/dt.
Linearity = additive + homogeneity = superposition. Zero input zero output is necessary condition but not suffice to prove linearity. System not satisfy zero input zero output system is non linear. Incrementally linear change in output is linearly related to the change in input. Can express it as a combo of linear system and adder that adds offset yzi(t) to output. Offset is zero input response . output can be determined by using the same linear combinations of initial outputs as the linear combo used to obtain x3 from x2 and x1. RLC in series y(t) across capacitor = 1/C (integrate i(tau) from inifinity to t). Memoryless systems are causal. Causality is a required condition for a system to physically realizable. Non causal system is predictive system and cannot be implemented. Class of systems with memory that require only a limited set of values of input x(t) in t0 T t t0 to compute the value of output y(t ). Such CT systems, whose response y(t ) is completely determined from the values of input x(t ) over the most recent past T time units, are referred to as finite-memory or Markov systems with memory of length T time unit. X(t/2) is non causal. Chapter 1 DRAW GRAPH PART Scale the signal x(t) by ||. The resulting waveform represents x(||t ).(ii) If is negative, invert the scaled signal x(||t ) with respect to the t = 0 axis. This step produces the waveform for x(t ). (iii) Shift the waveform for x(t ) obtained in step (ii) by |/| time units. Shift towards the right-hand side if (/) is negative. Otherwise, shift towards the left-hand side if (/) is positive. The waveform resulting from this step represents x(t + ), which is the required transformation.Chapter 3 representation of LTIC systems. For a time invariant
system coefficients are constant. RLC series circuit. X(t) is the input voltage. Output y(t) = zero input response + zero state response. Zero input is the component produced by the systems initial conditions and also called natural response. Zero input
is evaluated by solving a homogeneous eqn by setting x(t) = 0. Zero state response arises due to the input signal and does not depend on the initial conditions of the system. Initial conditions are assumed to be zero. Also called forced response of system. Defines steady state value of the output. Theorem 3.1 output of a first order differential equaltion, dy/dt + f(t) y(t) = r(t) resulting from input r(t) is given by y(t) = exp(-p) * [ integrate exp(p) r dt + C ], where p is given by p(t) = integral of f(t)dt and c is a constant. Delta function any signal x(t) can be represented as a linear combination of the time shifted impulse functions. Within a time interval of duration delta, say kdelta < t < (k + 1)delta, x(t ) is approximated by a constant value x(kdelta)_(t kdelta) delta. For a given value of t, say t = mdelta, only one term (k = m) on the right-hand side of Eq. (3.21) is non- zero. This is because only one of the shifted functions _(t kdelta) corresponding to k = m is non-zero. Which is same as which shows that a CT function can be represented as a weighted sumper imposition of the time shifted impulse function. Area enclosed by the unit impulse function equals unity. Impulse response The impulse response h(t ) of an LTIC system is the output of the system when a unit impulse (t ) is applied at the input. Because the system is LTIC, it satisfies the linearityand the time-shifting properties. If the input is a scaled and time-shifted impulse function a(t t0), the output, Eq. (3.25), of the system is also scaled by the factor of a and is time-shifted by t0, When an input signal x(t ) is passed through an LTIC system with impulse response h(t ), the resulting output y(t ) of the system can be calculated by convolving the input signal and the impulse response. Fourier basis functions If a complex exponential function is applied to an LTIC system with a real-valued impulse response function, the output response of the system is identical to the complex exponential function except for changes in amplitude and phase. In other words, output y(t) is given by where H(w1) = A1 exp(j phie1) is a complex valued constant for a given value of w1. A1 is the magnitude and phie1 is the phase of H(w. So output is given by convolution of input signal x(t) and impulse resp h(t). corollary output to a real valued impulse for a sin usodial function is another sinusoid with same frequency except for changes in phase and amplitude. Generalization of theorem input signal is periodic but is different from sinusoids or exponential we express input signal as a linear combinations of complex exponentials output ym(t) to the complex exp term xm(t) = km exp(j wm t) is given by using superposition, over all output is this is called exponential CTFS also can represent any arbitrary periodic signal as a linear combination of sinusoids Trignometric CTFS a0 represents mean or average value cosine terms represent even component of the zero mean signal x(t)-a0 also written as ev(x(t)-a0). CTFS coeff for symmetric signals
(i) If x(t ) is zero-mean, then a0 = 0.(ii) If x(t ) is an even function, then bn = 0 (iii) If x(t ) is a real function, then the trigonometric CTFS coefficients a0, an,and bn are also real-valued for all n (iv) If g(t ) = x(t ) + c (where c is a constant) then the trigonometric DTFS coefficients {ag0 , agn , bgn } of function g(t ) are related to the CTFS coefficients. Exponential Fourier Series Arbitrary periodicfunction with fundamental period T0 can be expressed as x(t) = = the second summation term is changed so that we get.Dn=a0forn=0,(anjbn)forn>0,(an+jbn)forn0, D-n = (an+jbn) for n>0. For real valued functions, an and bn are always real. Corollary magnitude and phase of exponential coefficients = |D-n| = |Dn| = underroot(an^2+bn^2). Phase D-n = - phase Dn = tan inverse (bn/an). Mag spectrum is even function and phase spectrum is odd. Parsevals Theorem power of a periodic signal can be calculated from its exponential CTFS coefficients For real valued signals |Dn| = |D-n| which gives us Linearity property exponentialcoeff. Of a linear combo of 2 periodic signals both having same fund. Period T0, are Given by the same linear combo of the exponential CTFS coeff for x1(t) and x2(t). Time shifting property if periodic signal is time shifted, the amplitude remains the same, phase changes by an exponential phase shift. . Phase changes from Dn by factor of nw0. Time Reversal If a periodic signal x(t ) is time-reversed, the amplitude spectrum remains unchanged. The phase spectrum changes by an exponential phase shift. if a signal is time-reversed, the CTFS coefficients of a time-reversed signal are the time-reversed CTFS coefficients of the original signal. . P(t) can be repped as exp CTFS
Time Scaling if a periodic signal with period T0 is time scaled, the CTFS spectra are inversely time scaled. Period of time scaled signal x(t/a) is given by T0/a. we get , r(t)=2(x(2t)) Differentiation and Integration exp CTFS coefficients of time differentiated and time integrated signals are expressed in terms of the exp CTFS coefficients of the original signal. Signal obtained by diff or integ a periodic signal over period T0 has same period T0 as original signal.
Integ(lnx)=xln(x)x
