Lab 4: First/Second Order DT Systems and a Communications Example (Second Draft)

Size: px
Start display at page:

Download "Lab 4: First/Second Order DT Systems and a Communications Example (Second Draft)"

Transcription

1 ECEN 33 Linear Systems Spring 3-- P. Mathys Lab 4: First/Second Order DT Systems and a Communications Example (Second Draft Introduction The main components from which linear and time-invariant discrete-time (LTI DT systems are made are adders, gains, and delay or memory elements. DT systems use DT sequences x n as inputs and produce DT sequences y n as outputs. The index n is an integer and x n, y n, and the state of a DT system are undefined when n is not an integer value. In practice DT signals are very often obtained by sampling continuous-time (CT signals with a sampling rate F s = /T s samples per second or hertz (Hz. Time t in seconds is then related to the index values n by t = n T s = n/f s. DT systems are typically implemented using digital computers in the form of microprocessors or DSP (digital signal processing chips which are essentially computers whose architecture has been optimized for DT signal processing.. Difference Equations Consider the first order CT lowpass filter (LPF with system function H L (s = Y (s X(s = ω L s + ω L. In the time domain this is described by the differential equation y ( (t + ω L y(t = ω L x(t, where x(t is the input and y(t is the output of the LPF. One way to convert this to a discrete time (DT system is to approximate the derivative by y ( (t y(t y(t T s T s, and to evaluate the differential equation at discrete time instants t = nt s, i.e., y ( (nt s + ω L y(nt s = ω L x(nt s = y(nt s y(nt s T s T s + ω L y(nt s = ω L x(nt s, where n is an integer and F s = /T s is the sampling rate in samples per second. Replacing y(nt s by y n and x(nt s by x n yields y n y n T s + ω L y n = ω L x n = y n y n + Ω L y n = Ω L x n,

2 where ω L T s = ω L /F s = Ω L is normalized frequency in radians. Thus, the system described by the first order difference equation ( + Ω L y n y n = Ω L x n, can be interpreted as a DT version of a first order LPF. A block diagram, using an adder, gains, and a delay cell D, that implements this difference equation is shown below. y n x n + Ω L + + y n +Ω L D y n More generally, an N-th order DT system is described by a difference equation that relates the output y n and its shifts y n,..., y n N to the input x n and its shifts x n,..., x n N. To be able to work conveniently with difference equations and to obtain system functions for DT systems, the next step is to use a transformation that converts time shifts into multiplicative operators.. z-transform The unilateral or one-sided z-transform X(z of a DT sequence x n, n, is defined as X(z = x n z n, n= and if x n < r for all n > n, then X(z converges for all values of z in the complex z-plane outside a circle of radius r. Example: DT Exponential. The DT exponential function a n u n has z-transform a n z n = n= a z, if a z <. The region of convergence (ROC of this z-transform is z > a, i.e., the whole z-plane outside a circle of radius a. Note that if a = then the transform pair u n is obtained. A related z-transform pair is δ n z δ n z n = n=

3 where the DT impulse δ n = if and only if n =. Example: DT Sinusoids. The complex-valued DT exponential function e ±jω n u n has z-transform e ±jω n z n =, ROC: z >. e ±jω z Using Euler s relation n= cos(ω n u n = ejω n + e jω n u n cos(ω z cos(ω z + z and sin(ω n u n = ejω n e jω n j u n sin(ω z cos(ω z + z z-transform Properties. Linearity. The z-transform is linear, i.e., α x [n] + β x [n] α X (z + β X (z, for any (real or complex valued constants α and β. Time Shift. If x n X(z, then x n has z-transform x n z n = x + z n= n= x n z (n, and thus x n z X(z + x, where z is recognized as a multiplicative delay operator..3 First Order DT Systems The difference equation of a general first order DT system is of the form y n + a y n = b x n + b x n. Using the one-sided z-transform and assuming zero intial conditions, this yields ( + a z Y (z = (b + b z X(z = H(z = Y (z X(z = b + b z + a z. - To be completed - 3

4 .4 Second Order DT Systems A general second order DT system is characterized in the time domain by a difference equation of the form In the z-domain this translates into and thus y n + a y n + a y n = b x n + b x n + b x n. ( + a z + a z Y (z = (b + b z + b z X(z, if zero initial conditions are assumed. - To be completed - H(z = Y (z X(z = b + b z + b z + a z + a z,.5 Frequency Response of DT Systems - To be completed -.6 First Order DT LPF The system function of a first order DT LPF with a zero at z = and a pole at z = r is The frequency response of this system is H L (z = K + z r z. H L (e jω = K + e jω + cos Ω j sin Ω = K r e jω r cos Ω + j r sin Ω = K + e jω r e jω ( r( + cos Ω j( + r sin Ω = K, r e jω r ejω r cos Ω + r where the first form is more suitable for the computation of H L (e jω and the second form is better suited for the computation of H L (e jω. Note that H L (e j = K r = select K = r for a dc gain of. With this value of K the magnitude squared of the frequency response is H L (e jω = ( r 4 ( + cos Ω + sin Ω ( r cos Ω + r sin Ω = ( r 4 ( + cos Ω + r r cos Ω. 4

5 The -3 db (or half power frequency Ω 3 is thus defined implicitly as H L (e jω 3 = ( r 4 ( + cos Ω 3 + r r cos Ω 3 =. From this it follows that ( r ( + cos Ω 3 = + r r cos Ω 3 = cos Ω 3 = r + r. Using the trigonometric identity this can be converted to tan ( α ( tan Ω3 = cos Ω 3 = + r r + cos Ω 3 + r + r = cos α + cos α, ( ( r r = = Ω ( + r 3 = tan. + r Another frequency of interest is Ω 45, the frequency at which the phase response H(e jω is 45. The phase response of a first order DT LPF is (from the second form of H(e jω given above ( H(e jω = tan ( + r sin Ω. ( r( + cos Ω Therefore, Ω 45 is defined by ( + r sin Ω 45 ( r( + cos Ω 45 = = r + r = sin Ω 45 = tan + cos Ω 45 where the last equality is the trigonometric identity ( α tan = sin α + cos α. ( Ω45 The interpretation of the result is that Ω 3 = Ω 45 for a first order DT LPF, which is the same property that a first order CT LPF has.,.7 Second Order DT LPF A second order DT LPF with poles at r e ±jθ, r <, has system function H L (z = K ( + z ( r e jθ z ( r e jθ z = K ( + z r cos θ z + r z. Note that H L ( = K 4 r cos θ + r, and H L( =. 5

6 For a dc gain of, set K to K = r cos θ + r 4 = H L (z = Multiplying top and bottom by z yields r cos θ + r 4 ( + z r cos θ z + r z. H L (z = r cos θ + r 4 The frequency response of this system is z + + z z r cos θ + r z. H L (e jω = r cos θ + r 4 e jω + + e jω e jω r cos θ + r e jω. Use Euler s relation for e ±jω to obtain H L (e jω = r cos θ + r 4 ( + cos Ω ( + r cos Ω r cos θ + j ( r sin Ω. Thus, the magnitude squared of the frequency response is H L (e jω = ( r cos θ + r 6 and the phase of the frequency response is 4 ( + cos Ω ( ( + r cos Ω r cos θ + ( r sin Ω, ( H L (e jω = tan ( r sin Ω. ( + r cos Ω r cos θ To determine the normalized natural frequency Ω, which is defined as the frequency at which the phase of the LPF is 9, look at This is satisfied if ( H L (e jω = tan ( r sin Ω = 9. ( + r cos Ω r cos θ ( + r cos Ω r cos θ = = cos Ω = r cos θ. + r Using the trigonometric formula tan ( α this result can be reformulated as ( tan Ω = cos α + cos α = sin α ( cos α = ( + cos α sin, α = cos Ω + cos Ω = + r r cos θ + r + r cos θ, 6

7 and thus ( Ω = tan + r r cos θ. + r + r cos θ The frequency response at Ω is H L (e jω = = = = r cos θ + r ( + cos Ω 4 ( + r cos Ω r cos θ +j( r }{{} sin Ω = r cos θ + r + cos Ω r cos θ + ( r = tan ( Ω j ( r sin Ω j ( r r cos θ + r + r + r cos θ ( + j ( r + r r cos θ = r r cos θ( + r + r cos θ j ( r ( + r 4r cos θ. j ( r For a nd order CT LPF H L (jω = /(jζ. Thus, for a nd order DT LPF, the damping ratio ζ can be defined as H L (e jω = ( + r 4r cos θ j ( r = j ζ = ζ = r ( + r 4r cos θ..8 Second Order DT BPF Consider the following system function of a second order DT BPF H B (z = K ( z ( + z ( r e jθ z ( r e jθ z = K z r cos θ z + r z, with poles at z = re ±jθ and zeros at z = ±. Multiply top and bottom by z to obtain H B (z = K The frequency response can then be written as H B (e jω = K e jω e jω e jω r cos θ + r e jω = K z z z r cos θ + r z. Thus, the phase of the frequency response of the BPF is j sin Ω ( + r cos Ω r cos θ + j( r sin Ω. ( H B (e jω = 9 tan ( r sin Ω. ( + r cos Ω r cos θ 7

8 For a BPF the normalized natural frequency Ω is defined as the frequency at which H(e jω =. This occurs when the argument of the inverse tangent becomes infinity and therefore ( + r cos Ω r cos θ = = cos Ω = r + r cos θ. Using the trigonometric identity for tan(α/ this can also be written as ( tan Ω = cos Ω = + r r cos θ + cos Ω + r + r cos θ ( = Ω = tan + r r cos θ, + r + r cos θ which is the same result as the one found for the nd order DT LPF. Using ( + r cos Ω r cos θ = in H B (e jω at Ω = Ω yields H B (e jω j sin Ω = K ( + r = K cos Ω r cos θ +j( r }{{} sin Ω = Since Ω is also the center frequency of a second order BPF, the choice K = r adjusts the maximum gain of the BPF to at the center frequency Ω., j sin Ω = K j( r sin Ω r..9 Frequency Response Measurement for DT Systems The simplest way to measure the magnitude of the frequency response of a DT system in Simulink is to use a sweeping sinusoid ( chirp signal with a desired frequency range as input signal and then to display the absolute value (or the logarithm of the absolute value of the signal at the output of the DT system. Consider the following first order system. 8

9 To generate a DT sinusoidal input signal x n with a sweeping frequency from. to π rad on a logarithmic scale, the following Matlab code can be used: WW =.; %Normalized start frequency [rad] WW = pi; %Normalized end frequency [rad] nlen = ; %# of samples for chirp nn = [:nlen]; %Index n axis WW = logspace(log(ww,log(ww,length(nn; %Log-spaced W axis theta = cumsum(ww; %Phase theta[n] for chirp signal xn = cos(theta; %Sweeping cosine (chirp signal To run the Simulink model with this input for the case when H(z = K the Matlab code shown next is added: + z a z, with K = a, and a =.8, a =.8; %Pole location b = (-a/*[ ]; %Numerator of H(z a = [ -a]; %Denominator of H(z t = nn ; %Input for Simulink model u = xn ; sim( DTsys,[t( t(end] %Run Simulink model %Fixed step (size, discrete solver yn = yout ; %Sinusoidal response of model The Configuration Parameters used for running the Simulink model are: Solver Type: Fixed-step Solver: Discrete (no continuous states Fixed-step size: Input: [t,u], checked Limit data points to last:, unchecked A plot of log ( H(e jω versus Ω on a logarithmic scale is shown below. Frequency Response of DT System, b=[.,.], a=[,.8] log H(e jω [db] Ω [rad] (log scale 9

10 When it is desired to use a linear Ω axis, the chirp signal can be generated as WW = ; %Normalized start frequency [rad] WW = pi; %Normalized end frequency [rad] nlen = ; %# of samples for chirp nn = [:nlen]; %Index axis WW = linspace(ww,ww,length(nn; %Lin-spaced W axis theta = cumsum(ww; %Phase theta[n] for chirp signal xn = cos(theta; %Sweeping cosine (chirp signal The graph below shows H(e jω versus a linear Ω/π axis for the same DT system as before. Frequency Response of DT System, b=[.,.], a=[,.8].8 H(e jω Ω/π. Step-Invariant CT to DT Conversion If a CT system is replaced by a DT system, not all features of the CT system (even if it is bandlimited can be preserved. If it is important to duplicate the unit step response as closely as possible, then a step-invariant CT to DT conversion can be used. Starting from the CT system function H(s one can obtain G(s and, using the inverse Laplace transform, the CT unit step response g(t. Using a sampling rate F s = /T s, the DT unit step response then becomes g n = g(n T s. The z-transform of g n is G(z which then leads to the DT system function as H(z = ( z G(z. Example: Step-Invariant st Order LPF CT to DT Conversion. Start from the CT LPF H(s = ω L ω L = G(s = s+ω L s (s+ω L = s = g(t = ( e ω Lt u(t. s+ω L Next, sampling g(t at t = n T s yields g n = g(n T s = ( e Ω Ln u n = G(z = z e = ( e ΩL z Ω L z ( z ( e Ω L z,

11 where Ω L = ω L /F s. From this the DT system function is obtained as H(z = ( z G(z = ( e Ω L z e Ω L z = e ΩL z e Ω L Thus, H(z has a pole p = e Ω L, a zero at infinity, and. H( = (dc gain, and H( = e Ω L + e Ω L (gain at Ω = π. Graphs of g(t and g n = g(nt s are shown in the figure below for ω L = 5 rad/sec and F s = Hz. Step Invariant LPF CT > DT, ω L =5, F s = Hz, F ss = Hz g(t, g n =g(nt s CT: g(t DT: g(n/fs t [sec] The sampling rate F ss is the rate that was used to simulate the CT model in Simulink. As expected, the unit step response of the DT and the CT systems are identical at the sampling time instants. But for the magnitude and the phase of the frequency response shown below, discrepancies between the DT and the CT system are clearly visible.

12 H(jω, H(e jω/fs Step Invariant LPF CT > DT, ω L =5, F s = Hz CT: H(jω DT: H(e jω/fs ω [rad/sec] H(jω, H(e jω/fs [deg] CT: H(jω DT: H(e jω/fs ω [rad/sec] Note that the frequency response plots were generated using the freqs (for the CT system and the freqz (for the DT system commands in Matlab. Example: Step-Invariant nd Order LPF CT to DT Conversion. A second order CT LPF with natural frequency ω and damping ratio ζ, < ζ, has system function H(s = ω s + ζω s + ω = ω (s + α jβ(s + α + jβ, where α = ζω and β = (ζ. Note that α +β = ω. The unit step response is obtained from ω G(s = s (s + α jβ(s + α + jβ = A s + B s + α jβ + B s + α + jβ, as g(t = A + B e αt e jβt + B e αt e jβt, for t >, where A = and ω B = = s(s+α+jβ s= α+jβ ω ( α+jβjβ = ω /(β β + jα Note that B = ω β = ( α ζ, and B = tan β jα = β β = + jα β. ( α = π tan. β

13 Sampling g(t at t = nt s, where F s = /T s is the sampling rate, yields g(nt s = A + B e αnts e jβnts + B e αnts e jβnts, for n. Define g n = g(nt s, Ω α = α = ζω F s F s, Ω β = β = F s and rewrite the sampled step response as ζ ω F s, The z-transform of this is g n = A + B e Ωαn e jω βn + B e Ωαn e jω βn, for n. G(z = A z + B e Ωα e + B jω β z e Ωα e. jω β z To obtain the DT system function H(z, multiply by z : H(z = A + B( z e Ωα e jω β z + B ( z e Ωα e jω β z = b + b z + b z e Ωα cos Ω β z + e Ωα z, where b = A + B + B =, b = e Ωα ( A cos Ωβ + B cos( Ω β + B Re{B}, b = Ae Ωα + B e Ωα cos( Ω β + B. Plots of g(t and g n = g(nt s are shown in the following figure for ω = 5 rad/sec, ζ = /, when F s = Hz..4 Step Invariant LPF Order CT > DT, ω =5, ζ=.5, F s = Hz, F ss = Hz. g(t, g n =g(nt s CT: g(t DT: g(n/fs t [sec] As can be seen, the step responses of the CT and DT systems are identical at the sampling time instants. But the next two plots show that the magnitude and the phase responses of the CT and DT systems differ, especially near ω = π F s. 3

14 Step Invariant LPF Order CT > DT, ω =5, ζ=.5, F s = Hz H(jω, H(e jω/fs 3 4 CT: H(jω DT: H(e jω/fs ω [rad/sec] H(jω, H(e jω/fs [deg] 5 5 CT: H(jω DT: H(e jω/fs ω [rad/sec]. Step Response Measurement for DT Systems To generate and display the step response g n of a DT system in Simulink, start from generating an index axis nn and a unit step un as shown below. nlen = 5; nstep = 5; nn = [:nlen]; un = zeros(size(nn; ix = find(nn>=nstep; un(ix = ones(size(ix; %# of samples for g[n] %Step index %Index axis for g[n] %Prepare unit step %Unit step starts at nstep %DT unit step signal Note that nlen and nstep may have to be adjusted depending on the properties of the system to be measured. To obtain the step response of a first order DT system with H(z = K the following Matlab code can be used: + z a z, with K = a, and a =.8, 4

15 a =.8; %Pole location b = (-a/*[ ]; %Numerator of H(z a = [ -a]; %Denominator of H(z t = nn ; %Input for Simulink model u = un ; sim( DTsys,[t( t(end] %Run Simulink model %Fixed step (size, discrete solver gn = yout ; %Step response The Configuration Parameters used for running the Simulink model are: Solver Type: Fixed-step Solver: Discrete (no continuous states Fixed-step size: Input: [t,u], checked Limit data points to last:, unchecked The graph below shows the unit step response of the DT system. Step Response of DT System, b=[.,.], a=[,.8].8 g[n] n The command that was used to produce the stem plot in Matlab is stem(nn,gn,.-r. Application: FSK Data Transmission Frequency shift keying (FSK is a data transmission method for digital data that uses sinusoids with different frequencies to transmit different data values. The simplest case is binary FSK which uses two different frequencies, f to transmit binary or space (S and f to transmit binary or mark (M. One way to describe a binary FSK signal s(t is as follows. Let d n be the binary data sequence that is to be transmitted and define d [n] = {, if dn =,, if d n =, d [n] = d n. 5

16 Then use the DT sequences d [n] and d [n] to define the two CT data waveforms d (t = d [n] p(t nt B, and d (t = n= d [n] p(t nt B, where F B = /T B is the symbol or baud rate at which the data is transmitted and p(t is a rectangular pulse of width T B and amplitude, i.e., {, if t < TB, p(t =, otherwise. The next step is to convert d (t and d (t to sinusoidal on-off signals s (t and s (t with frequencies f and f by setting s (t = d (t cos(πf t + θ, and s (t = d (t cos(πf t + θ. In general, the phases θ and θ of s (t and s (t could be fixed or time varying and may or may not be known at the receiving end. Finally, the complete binary FSK signal is s(t = s (t + s (t = d (t cos(πf t + θ + d (t cos(πf t + θ. An example of an FSK signal with F B = baud, f = 3 Hz, f = Hz is shown below for the data sequence d n = [,,,,,,,,, ]. n= FSK Example: f =3 Hz, f = Hz, F B = Baud, d n =[,,,,,,,,,] d (t, s (t d (t, s (t d (t, s(t t [sec] 6

17 The first graph shows d (t (dashed, green and s (t (solid, blue and the second graph shows d (t (dashed, green and s (t (solid, blue. The third graph shows the FSK signal s(t that actually transmits the data sequence d n, and the data pattern is easy to see directly by visual inspection. Let r(t = γs(t, where γ represents the attenuation of the transmission channel, be a received binary FSK signal. The goal of an FSK receiver is to recover the transmitted data sequence d n, despite the unknown channel attenuation, phase and frequency errors, and noise picked up during the transmission. Conceptually, the simplest way to do this is is to use two bandpass filters (BPF, one centered at f and one centered at f. At suitably chosen sampling time instants T B seconds apart, the average power (averaged over T B seconds at the outputs of the BPFs is compared and d n is set to one if the average power at f is larger than the average power at f ; otherwise d n is set to zero. Another approach is to use the fact that the product of two sinusoids with frequencies f a and f b yields the sum of two sinusoids, one at the difference frequency f a f b and the other one at the sum frequency f a + f b. In this way the signal s (t can be shifted down to dc (or baseband by multiplication with a sinusoid with frequency f, and s (t can be shifted down to dc (or baseband by multiplication with a sinusoid with frequency f. One slight problem is that in general the phases of the sinusoids that were used at the transmitter are not known at the receiver. This problem can be solved by multiplying the received signal with both a sine and a cosine at the frequencies f and f at the receiver. The block diagram of a FSK receiver that uses this strategy is shown in the figure below. v i (t cosπf t v q (t LPF v Li (t at f L (. LPF v Lq (t at f L ( ˆd (t r(t sinπf t v i (t cosπf t v q (t LPF v Li (t at f L (. LPF v Lq (t at f L ( ˆd (t sinπf t Using the trigonometric identities cos α cos β = [ ] [ ] cos(α β+cos(α+β, and cos α sin β = sin(α β sin(α+β, 7

18 the quantity v i (t after multiplication of r(t with cos πf t is computed as v i (t = r(t cos πf t = γ ( d (t cos(πf t + θ + d (t cos(πf t + θ cos πf t = γd (t cos θ + γd (t cos(4πf t + θ + +γd (t cos(π(f f t + θ + γd (t cos(π(f + f t + θ. The main term of interest is the underlined γd (t cos θ. All other terms are at higher frequencies, like f, and f ± f. Thus, if the LPF at f L is designed to reject these frequencies, then v Li = γd (t cos θ. The quantity v q (t is similarly computed as v q (t = r(t sin πf t = γ ( d (t cos(πf t + θ + d (t cos(πf t + θ sin πf t = γd (t sin θ γd (t sin(4πf t + θ + +γd (t sin(π(f f t + θ γd (t sin(π(f + f t + θ. Except for the term of interest (underlined, all other terms are at higher frequencies, namely f, and f ±f. Thus, using an LPF that rejects these terms, results in v Lq = γd (t sin θ. But note that the LPFs need to have sufficient bandwidth to pass a data sequence of alternating s and s which has a fundamental frequency of F B /. The last step needed to obtain an estimate ˆd (t of d (t is ˆd (t = ( γd (t cos θ + ( γd (t sin θ = γ d (t. The quantities v Li (t and v Lq (t are obtained using entirely analogous computations as v Li (t = γd (t cos θ, and v Lq (t = γd (t sin θ, and thus the estimate ˆd (t of d (t is ˆd (t = ( γd (t cos θ + ( γd (t sin θ = γ d (t. From ˆd (t and ˆd (t the CT version ˆd(t of the received data sequence ˆd n can be obtained as ˆd(t = {, if ˆd (t ˆd (t,, otherwise, for any given value of t. Finally, ˆd n is recovered by sampling ˆd(t as ˆd n = ˆd(nT B + τ, where τ is a suitably chosen delay which compensates for the delay through the LPFs at the receiver. The next figure shows v i (t, v Li (t, v q (t, v Lq (t, v i (t, v Li (t, and v q (t, v Lq (t, for the sample FSK signal that was given earlier. The LPFs that were used to produce the graphs were nd order LPFs with ω = πf B / and ζ = /. 8

19 FSK Receiver: f =3 Hz, f = Hz, F B = Baud, d n =[,,,,,,,,,] v i (t, v Li (t v q (t, v Lq (t v i (t, v Li (t v q (t, v Lq (t t [sec] The next set of graphs shows ˆd (t, ˆd (t, ˆd(t, and the sampled values at times t = nt B + τ. The sampling delay τ is approximately equal to T B. FSK Receiver: f =3 Hz, f = Hz, F B = Baud, d n =[,,,,,,,,,] dhat (t dhat (t dhat(t t [sec] 9

20 From the samples in the last graph it is easy to see that the transmitted data sequence can be successfully recovered at the receiver. Prelab Questions P. Phase Shift of First Order DT Filters. Let H(z = K z z p z, be the system function of a first order DT system. Assuming that the system is BIBO stable and z <, determine for which values of p and z the phase H(e jω of the frequency response is > and for which it is <. P. Pole-Zero Specification of DT System. The following pole-zero plot defines a second order DT system with real poles at a and a. Im{z} <a< -a a Re{z} Determine the system function H(z such that max Ω H(e jω =. What type of filter is this? What is the unit step response g n of this system? P3. Step Invariant CT to DT System Conversion. (a Start from the CT system with system function H(s = s. s + ω x Determine the system function H(z of the corresponding step-invariant DT system. (b Repeat (a for the CT system with system function H(s = ζω s s + ζω s + ω. Assume that < ζ.

21 3 Lab Experiments E. Pole/Zero Placement for First and Second Order DT Systems. The goal of this experiment is to explore the relationship between pole/zero placement and the step and frequency responses of first and second order DT systems. Use the Simulink model shown below for first order DT systems. The Simulink model shown in the next figure is used to simulate second order DT systems. In both cases use the following Configuration Parameters for running the Simulink model: Solver Type: Fixed-step Solver: Discrete (no continuous states Fixed-step size:

22 Input: [t,u], checked Limit data points to last:, unchecked (a Let H(z be the first order system function H(z = K z z p z, with zero z and pole p. Make plots of the magnitude of the frequency response H(e jω, the unit step response g n, and the pole/zero locations for the following combinations of poles and zeros: p =.9,.3 and z =,, + (6 combinations. In each case choose K such that the passband gain (dc gain for LPF, center frequency gain for BPF, gain at highest frequency for HPF of H(z is. If the -3 db frequency is much smaller than π, display the magnitude of the frequency response in db versus Ω on a logarithmic scale, as shown in the example below. Frequency Response of DT System, b=[.5,], a=[,.95] log H(e jω [db] Ω 3 Ω [rad] (log scale Step Response Pole Zero Plot.8.5 g[n].6.4. Im{z} n.5.5 Re{z} If the -3 db frequency is close to or larger, then use a linear display for H(e jω as shown in the next example.

23 Frequency Response of DT System, b=[.45,.45], a=[,.].8 H(e jω Ω Ω [rad] Step Response Pole Zero Plot.8.5 g[n].6.4. Im{z} n.5.5 Re{z} For which values of z do you get LPFs? For which do you get HPFs? What happens if z = /p is selected? For which values of p can you use a straight line Bode plot approximation for H(e jω over at least one decade to the left and to the right of the -3dB frequency? (b Let H(z be the second order DT system function H(z = K ( z z ( z z ( re jθ z ( re jθ z, with zeros z, z, poles re ±jθ and gain factor K. Now there are essentially three quantities that determine the behavior of the system: The location of the zeros, often at z =, z =, or both, the magnitude r, and the angle θ of the complex conjugate poles. Consider the following three sets of zero locations: z = z =, z =, z =, and z = z =. Make plots of the magnitude of the frequency response H(e jω, the unit step response g n, and the pole/zero locations for the three sets of zero locations and the following r, θ combinations: (i r =.93, θ = 4. (ii r =.98, θ = 4. (iii r =.5, θ = 5. (iv r =.8, θ = 5. 3

24 In each case adjust the gain factor K so that the passband gain of the system is. What determines whether H(z is the system function of a LPF, BPF, or HPF? Try to determine the natural frequency Ω and the damping factor ζ for each of the cases above. Which parameters determine the values of Ω and ζ? Hint: In addition to using Simulink, consider also using the freqz command to plot the magnitude and the phase of the frequency response. Depending on the pole locations you may want to choose between a logarithmic or a linear display of the frequency response. For which of the above cases would a straight line Bode plot approximation be useful? For which would it not be useful? Here is an example of a logarithmic frequency response display when r =.98 and θ = 45. The purple lines show the sinusoidal output from the Simulink simulation with a chirp signal. The blue line was computed using the freqz command. Frequency Response of DT System, b=[.98,,.98], a=[,.39,.96] log H(e jω [db] Ω [rad] (log scale.5 Step Response Pole Zero Plot.5 g[n] Im{z} n.5.5 Re{z} The next figure shows the same system with a linear frequency response display for Ω =... π. 4

25 Frequency Response of DT System, b=[.98,,.98], a=[,.39,.96].8 H(e jω Ω Ω 3 Ω [rad].5 Step Response Pole Zero Plot.5 g[n] Im{z} n.5.5 Re{z} The lines at Ω 3 and Ω 3 show the locations of the lower and upper -3dB frequencies of this BPF. (c Suppose you would like to design a second order DT LPF with Ω =.5 radians and ζ = /. Find z, z, r, and θ of the corresponding system function. Determine K such that the dc gain is. How easy (or difficult is it to find these parameters by trial and error? the figure below shows H(e jω and g n of the desired LPF 5

26 .4 Frequency Response of DT System, with Ω =.5 and ζ=.5. H(e jω Ω [rad].4 Step Response g[n] n E. Comparison of CT Systems and their DT Replacements. Modern signal processing implementations use programmable digital hardware whenever possible. A task that is encountered frequently is to convert an idea or a concept or an existing implementation from continuous time to discrete time. Because of the periodic nature of the frequency response of DT systems, a perfect match between CT and DT systems in both the time and frequency domains is impossible for practical implementations. Starting from a given CT system, the goal of this experiment is to determine an approximate DT equivalent and to compare the performance of the two systems in terms of their unit step response and the magnitude and the phase of the frequency response. In Simulink, use the same DT models and Configuration Parameter settings as in experiment. For first and second order CT systems use the following CT models. 6

27 Because Simulink has to use discrete time instants for both continuous time and discrete time system simulations, two sampling frequencies have to be used. One, which will be denoted by F ss, is the sampling rate used to simulate the CT system. The other one, denoted by F s, is the sampling rate of the DT system that is supposed to replace the CT system. Typically, F ss F s, e.g., F ss =... F s. Here is some sample Matlab code that can be used as a starting point to generate the unit step responses g(t and g n : 7

28 Fs =... %Sampling rate DT system Fss = *Fs; %Sampling rate CT system bct =... %Numerator of H(s act =... %Denominator of H(s bdt =... %Numerator of H(z adt =... %Denominator of H(z nlen = ; %Number of samples for g[n] nstep = ; %Step index of u[n] un = zeros(size(nn; %Prepare DT unit step ix = find(nn>=nstep; un(ix = ones(size(ix; %DT unit step b = bdt; %DT model parameters a = adt; t = nn ; %Input for Simulink model u = un ; sim( DTsys,[t( t(end] %Run DT Simulink model %Fixed step ( discrete solver gn = yout ; %DT step response tlen = nlen/fs; %Duration of g(t in sec tstep = nstep/fs; %Step time of u(t tt = [:round(tlen*fss-]/fss; %Time axis for g(t ut = zeros(size(tt; %Prepare CT unit step ix = find(tt>tstep; ut(ix = ones(size(ix; %CT unit step b = bct; %CT model parameters a = act; t = tt ; %Input for Simulink model u = ut ; sim( CTsys,[t( t(end] %Run CT Simulink model %Fixed step (/Fss ode3 solver gt = yout ; %CT Step response The Configuration Parameters for running the CT Simulink model are: Solver Type: Fixed-step Solver: ode3 (Bogacki-Shampine Fixed-step size: /Fss Input: [t,u], checked Limit data points to last:, unchecked Here is an example of a step-invariant CT to DT conversion of a st order LPF with ω L = 5 rad/sec, F s = /T s = 8 Hz, and F ss = 8 Hz. The unit step responses g(t and g n are identical at times t = nt s. 8

29 Step Invariant LPF Order CT > DT, ω L =5, F s =8 Hz, F ss =8 Hz g(t, g n =g(nt s CT: g(t DT: g(n/fs.5.5 t [sec] x 3 The magnitude and the phase of the frequency responses of the CT and DT systems, computed using the freqs and freqz commands, respectively, are shown next. Step Invariant LPF Order CT > DT, ω L =5, F s =8 Hz H(jω, H(e jω/fs 5 5 CT: H(jω DT: H(e jω/fs ω [rad/sec] x 4 H(jω, H(e jω/fs [deg] 3 CT: H(jω DT: H(e jω/fs ω [rad/sec] x 4 There are clearly some differences here, especially for ω near πf s and beyond. Note that the unwrap command was used for the phase to remove phase jumps by ±36. It is instructive to compare this also with a DT LPF with system function H(z = r 9 + z r z.

30 This has a pole at r. But the main difference is that it has a zero at z =, whereas the step-invariant DT system has no finite zero. The graph below shows the unit step responses of the CT system and the two DT systems. The value of r for the nd DT system is r =.56. Step Invariant LPF Order CT > DT, ω L =5, F s =8 Hz, F ss =8 Hz g(t, g n =g(nt s CT: g(t. DT: g(n/fs DT: g(n/fs.5.5 t [sec] x 3 In the frequency response which is shown below for all three systems, the efect of placing a zero at z = for the second DT system is clearly visible. Step Invariant LPF Order CT > DT, ω L =5, F s =8 Hz 5 H(jω, H(e jω/fs 5 5 CT: H(jω DT: H(e jω/fs DT: H(e jω/fs ω [rad/sec] x 4 H(jω, H(e jω/fs [deg] CT: H(jω 3 DT: H(e jω/fs DT: H(e jω/fs ω [rad/sec] x 4 3

31 (a Replacement of first order CT LPF with system function H L (s = ω L s + ω L, by either a first order step-invariant DT system (DT or a first order DT system with a pole at z = r and a zero at z = (DT. Let ω L = rad/sec and determine the minimum sampling rate F s needed for both DT replacements if the following requirements are set: (i The unit step response g n has to satisfy g n g n. fo all n and g(nt s g n. for all n. Adjust the value of r for the second DT system (DT as necessary. (ii The phase of the frequency response has to satisfy H L (jω H L (e jω/fs 5 for ω ω L. Adjust the value of r for the second DT system (DT as necessary. (b Replacement of second order CT LPF with system function H L (s = ω s + ζω s + ω, by either a second order step-invariant DT system (DT or a second order DT system with poles at z = re jθ and a double zero at z = (DT. Let ω = and ζ =.46 and determine the minimum sampling rate F s needed for both DT replacements if the following conditions have to be met: (i The maximum overshoot of g n has to be between 8% and % and the -3 db frequency of the DT system has to be within ±5% of the -3 db frequency of the CT system. Adjust the values of r and θ for the second DT system (DT as necessary. (ii The maximum overshoot of g n has to be between 8% and % and the magnitude of the frequency response in decibel of the DT system can differ by at most db from the magnitude of the frequency response in decibel of the CT system for frequencies in the range ω =... rad/sec. Adjust the values of r and θ for the second DT system (DT as necessary. E3. Design of a Receiver for FSK Signals. The goal of this experiment is to demodulate binary FSK signals and extract the received ASCII text messages. The parameters used for the FSK signals are identical to the ones that are used for caller ID in the US. The baud (or symbol rate is baud. The frequency to transmit a binary (or space is f = Hz and the frequency to transmit a binary (or mark is f = Hz. A known test signal with the text The quick brown fox jumps over the lazy dog, is recorded in the wav file FSKsig.wav with a sampling rate F s = Hz. The ASCII text uses 8 bits per character and the parallel to serial conversion is done such that the LSB (least significant bit is transmitted first, in the same way as described for Lab. (a Write a Matlab script that reads an FSK signal from a wav file and processes it using the FSK receiver described in the introduction. For the LPFs design nd order DT LPFs 3

32 with natural frequencies (in Hz approximately equal to F B /. Choose a ζ such that the step response of the LPF has an overshoot of about... 5%. Implement the LPFs using Simulink. Use the test FSK signal in FSKsig.wav to test the correct operation of your receiver. (b Use the receiver you designaed in (a to extract the text message from the binary FSK signal in FSKsig.wav. c, P. Mathys. Last revised: 3-9-, PM. 3

Lab 1: First Order CT Systems, Blockdiagrams, Introduction

Lab 1: First Order CT Systems, Blockdiagrams, Introduction ECEN 3300 Linear Systems Spring 2010 1-18-10 P. Mathys Lab 1: First Order CT Systems, Blockdiagrams, Introduction to Simulink 1 Introduction Many continuous time (CT) systems of practical interest can

More information

The University of Texas at Austin Dept. of Electrical and Computer Engineering Final Exam

The University of Texas at Austin Dept. of Electrical and Computer Engineering Final Exam The University of Texas at Austin Dept. of Electrical and Computer Engineering Final Exam Date: December 18, 2017 Course: EE 313 Evans Name: Last, First The exam is scheduled to last three hours. Open

More information

Problem Set 1 (Solutions are due Mon )

Problem Set 1 (Solutions are due Mon ) ECEN 242 Wireless Electronics for Communication Spring 212 1-23-12 P. Mathys Problem Set 1 (Solutions are due Mon. 1-3-12) 1 Introduction The goals of this problem set are to use Matlab to generate and

More information

The University of Texas at Austin Dept. of Electrical and Computer Engineering Midterm #2

The University of Texas at Austin Dept. of Electrical and Computer Engineering Midterm #2 The University of Texas at Austin Dept. of Electrical and Computer Engineering Midterm #2 Date: November 18, 2010 Course: EE 313 Evans Name: Last, First The exam is scheduled to last 75 minutes. Open books

More information

Wireless Communication

Wireless Communication ECEN 242 Wireless Electronics for Communication Spring 22-3-2 P. Mathys Wireless Communication Brief History In 893 Nikola Tesla (Serbian-American, 856 943) gave lectures in Philadelphia before the Franklin

More information

2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal.

2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal. 1 2.1 BASIC CONCEPTS 2.1.1 Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal. 2 Time Scaling. Figure 2.4 Time scaling of a signal. 2.1.2 Classification of Signals

More information

Boise State University Department of Electrical and Computer Engineering ECE 212L Circuit Analysis and Design Lab

Boise State University Department of Electrical and Computer Engineering ECE 212L Circuit Analysis and Design Lab Objectives Boise State University Department of Electrical and Computer Engineering ECE L Circuit Analysis and Design Lab Experiment #0: Frequency esponse Measurements The objectives of this laboratory

More information

EELE 4310: Digital Signal Processing (DSP)

EELE 4310: Digital Signal Processing (DSP) EELE 4310: Digital Signal Processing (DSP) Chapter # 10 : Digital Filter Design (Part One) Spring, 2012/2013 EELE 4310: Digital Signal Processing (DSP) - Ch.10 Dr. Musbah Shaat 1 / 19 Outline 1 Introduction

More information

LECTURER NOTE SMJE3163 DSP

LECTURER NOTE SMJE3163 DSP LECTURER NOTE SMJE363 DSP (04/05-) ------------------------------------------------------------------------- Week3 IIR Filter Design -------------------------------------------------------------------------

More information

Sinusoids. Lecture #2 Chapter 2. BME 310 Biomedical Computing - J.Schesser

Sinusoids. Lecture #2 Chapter 2. BME 310 Biomedical Computing - J.Schesser Sinusoids Lecture # Chapter BME 30 Biomedical Computing - 8 What Is this Course All About? To Gain an Appreciation of the Various Types of Signals and Systems To Analyze The Various Types of Systems To

More information

Outline. EECS 3213 Fall Sebastian Magierowski York University. Review Passband Modulation. Constellations ASK, FSK, PSK.

Outline. EECS 3213 Fall Sebastian Magierowski York University. Review Passband Modulation. Constellations ASK, FSK, PSK. EECS 3213 Fall 2014 L12: Modulation Sebastian Magierowski York University 1 Outline Review Passband Modulation ASK, FSK, PSK Constellations 2 1 Underlying Idea Attempting to send a sequence of digits through

More information

Basic Signals and Systems

Basic Signals and Systems Chapter 2 Basic Signals and Systems A large part of this chapter is taken from: C.S. Burrus, J.H. McClellan, A.V. Oppenheim, T.W. Parks, R.W. Schafer, and H. W. Schüssler: Computer-based exercises for

More information

Project 2 - Speech Detection with FIR Filters

Project 2 - Speech Detection with FIR Filters Project 2 - Speech Detection with FIR Filters ECE505, Fall 2015 EECS, University of Tennessee (Due 10/30) 1 Objective The project introduces a practical application where sinusoidal signals are used to

More information

Digital Video and Audio Processing. Winter term 2002/ 2003 Computer-based exercises

Digital Video and Audio Processing. Winter term 2002/ 2003 Computer-based exercises Digital Video and Audio Processing Winter term 2002/ 2003 Computer-based exercises Rudolf Mester Institut für Angewandte Physik Johann Wolfgang Goethe-Universität Frankfurt am Main 6th November 2002 Chapter

More information

NH 67, Karur Trichy Highways, Puliyur C.F, Karur District DEPARTMENT OF INFORMATION TECHNOLOGY DIGITAL SIGNAL PROCESSING UNIT 3

NH 67, Karur Trichy Highways, Puliyur C.F, Karur District DEPARTMENT OF INFORMATION TECHNOLOGY DIGITAL SIGNAL PROCESSING UNIT 3 NH 67, Karur Trichy Highways, Puliyur C.F, 639 114 Karur District DEPARTMENT OF INFORMATION TECHNOLOGY DIGITAL SIGNAL PROCESSING UNIT 3 IIR FILTER DESIGN Structure of IIR System design of Discrete time

More information

Final Exam Solutions June 14, 2006

Final Exam Solutions June 14, 2006 Name or 6-Digit Code: PSU Student ID Number: Final Exam Solutions June 14, 2006 ECE 223: Signals & Systems II Dr. McNames Keep your exam flat during the entire exam. If you have to leave the exam temporarily,

More information

Project I: Phase Tracking and Baud Timing Correction Systems

Project I: Phase Tracking and Baud Timing Correction Systems Project I: Phase Tracking and Baud Timing Correction Systems ECES 631, Prof. John MacLaren Walsh, Ph. D. 1 Purpose In this lab you will encounter the utility of the fundamental Fourier and z-transform

More information

Lecture 17 z-transforms 2

Lecture 17 z-transforms 2 Lecture 17 z-transforms 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/5/3 1 Factoring z-polynomials We can also factor z-transform polynomials to break down a large system into

More information

CHAPTER 6 INTRODUCTION TO SYSTEM IDENTIFICATION

CHAPTER 6 INTRODUCTION TO SYSTEM IDENTIFICATION CHAPTER 6 INTRODUCTION TO SYSTEM IDENTIFICATION Broadly speaking, system identification is the art and science of using measurements obtained from a system to characterize the system. The characterization

More information

Lecture 2 Review of Signals and Systems: Part 1. EE4900/EE6720 Digital Communications

Lecture 2 Review of Signals and Systems: Part 1. EE4900/EE6720 Digital Communications EE4900/EE6420: Digital Communications 1 Lecture 2 Review of Signals and Systems: Part 1 Block Diagrams of Communication System Digital Communication System 2 Informatio n (sound, video, text, data, ) Transducer

More information

1. Clearly circle one answer for each part.

1. Clearly circle one answer for each part. TB 1-9 / Exam Style Questions 1 EXAM STYLE QUESTIONS Covering Chapters 1-9 of Telecommunication Breakdown 1. Clearly circle one answer for each part. (a) TRUE or FALSE: Absolute bandwidth is never less

More information

Experiments #6. Convolution and Linear Time Invariant Systems

Experiments #6. Convolution and Linear Time Invariant Systems Experiments #6 Convolution and Linear Time Invariant Systems 1) Introduction: In this lab we will explain how to use computer programs to perform a convolution operation on continuous time systems and

More information

ECE503: Digital Filter Design Lecture 9

ECE503: Digital Filter Design Lecture 9 ECE503: Digital Filter Design Lecture 9 D. Richard Brown III WPI 26-March-2012 WPI D. Richard Brown III 26-March-2012 1 / 33 Lecture 9 Topics Within the broad topic of digital filter design, we are going

More information

Fourier Transform Analysis of Signals and Systems

Fourier Transform Analysis of Signals and Systems Fourier Transform Analysis of Signals and Systems Ideal Filters Filters separate what is desired from what is not desired In the signals and systems context a filter separates signals in one frequency

More information

Signals and Systems Lecture 6: Fourier Applications

Signals and Systems Lecture 6: Fourier Applications Signals and Systems Lecture 6: Fourier Applications Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Winter 2012 arzaneh Abdollahi Signal and Systems Lecture 6

More information

Spring 2018 EE 445S Real-Time Digital Signal Processing Laboratory Prof. Evans. Homework #1 Sinusoids, Transforms and Transfer Functions

Spring 2018 EE 445S Real-Time Digital Signal Processing Laboratory Prof. Evans. Homework #1 Sinusoids, Transforms and Transfer Functions Spring 2018 EE 445S Real-Time Digital Signal Processing Laboratory Prof. Homework #1 Sinusoids, Transforms and Transfer Functions Assigned on Friday, February 2, 2018 Due on Friday, February 9, 2018, by

More information

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING. ECE 2025 Fall 1999 Lab #7: Frequency Response & Bandpass Filters

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING. ECE 2025 Fall 1999 Lab #7: Frequency Response & Bandpass Filters GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Fall 1999 Lab #7: Frequency Response & Bandpass Filters Date: 12 18 Oct 1999 This is the official Lab #7 description;

More information

DSP First Lab 08: Frequency Response: Bandpass and Nulling Filters

DSP First Lab 08: Frequency Response: Bandpass and Nulling Filters DSP First Lab 08: Frequency Response: Bandpass and Nulling Filters Pre-Lab and Warm-Up: You should read at least the Pre-Lab and Warm-up sections of this lab assignment and go over all exercises in the

More information

Frequency Response Analysis

Frequency Response Analysis Frequency Response Analysis Continuous Time * M. J. Roberts - All Rights Reserved 2 Frequency Response * M. J. Roberts - All Rights Reserved 3 Lowpass Filter H( s) = ω c s + ω c H( jω ) = ω c jω + ω c

More information

SECTION 7: FREQUENCY DOMAIN ANALYSIS. MAE 3401 Modeling and Simulation

SECTION 7: FREQUENCY DOMAIN ANALYSIS. MAE 3401 Modeling and Simulation SECTION 7: FREQUENCY DOMAIN ANALYSIS MAE 3401 Modeling and Simulation 2 Response to Sinusoidal Inputs Frequency Domain Analysis Introduction 3 We ve looked at system impulse and step responses Also interested

More information

Fall Music 320A Homework #2 Sinusoids, Complex Sinusoids 145 points Theory and Lab Problems Due Thursday 10/11/2018 before class

Fall Music 320A Homework #2 Sinusoids, Complex Sinusoids 145 points Theory and Lab Problems Due Thursday 10/11/2018 before class Fall 2018 2019 Music 320A Homework #2 Sinusoids, Complex Sinusoids 145 points Theory and Lab Problems Due Thursday 10/11/2018 before class Theory Problems 1. 15 pts) [Sinusoids] Define xt) as xt) = 2sin

More information

Digital Processing of Continuous-Time Signals

Digital Processing of Continuous-Time Signals Chapter 4 Digital Processing of Continuous-Time Signals 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 4-1-1 Digital Processing of Continuous-Time Signals Digital

More information

Digital Processing of

Digital Processing of Chapter 4 Digital Processing of Continuous-Time Signals 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 4-1-1 Digital Processing of Continuous-Time Signals Digital

More information

ECE438 - Laboratory 7a: Digital Filter Design (Week 1) By Prof. Charles Bouman and Prof. Mireille Boutin Fall 2015

ECE438 - Laboratory 7a: Digital Filter Design (Week 1) By Prof. Charles Bouman and Prof. Mireille Boutin Fall 2015 Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 7a: Digital Filter Design (Week 1) By Prof. Charles Bouman and Prof. Mireille Boutin Fall 2015 1 Introduction

More information

Signals and Systems Lecture 6: Fourier Applications

Signals and Systems Lecture 6: Fourier Applications Signals and Systems Lecture 6: Fourier Applications Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Winter 2012 arzaneh Abdollahi Signal and Systems Lecture 6

More information

Lab 10: Phase and Hybrid Amplitude/Phase Shift Keying, Carrier Sync

Lab 10: Phase and Hybrid Amplitude/Phase Shift Keying, Carrier Sync ECEN 4652/5002 Communications Lab Spring 2017 04-24-17 P. Mathys Lab 10: Phase and Hybrid Amplitude/Phase Shift Keying, Carrier Sync 1 Introduction A sinusoid like A cos(2πft + θ is characterized by its

More information

Infinite Impulse Response Filters

Infinite Impulse Response Filters 6 Infinite Impulse Response Filters Ren Zhou In this chapter we introduce the analysis and design of infinite impulse response (IIR) digital filters that have the potential of sharp rolloffs (Tompkins

More information

PULSE SHAPING AND RECEIVE FILTERING

PULSE SHAPING AND RECEIVE FILTERING PULSE SHAPING AND RECEIVE FILTERING Pulse and Pulse Amplitude Modulated Message Spectrum Eye Diagram Nyquist Pulses Matched Filtering Matched, Nyquist Transmit and Receive Filter Combination adaptive components

More information

Lecture 3 Complex Exponential Signals

Lecture 3 Complex Exponential Signals Lecture 3 Complex Exponential Signals Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/3/1 1 Review of Complex Numbers Using Euler s famous formula for the complex exponential The

More information

ECE 203 LAB 2 PRACTICAL FILTER DESIGN & IMPLEMENTATION

ECE 203 LAB 2 PRACTICAL FILTER DESIGN & IMPLEMENTATION Version 1. 1 of 7 ECE 03 LAB PRACTICAL FILTER DESIGN & IMPLEMENTATION BEFORE YOU BEGIN PREREQUISITE LABS ECE 01 Labs ECE 0 Advanced MATLAB ECE 03 MATLAB Signals & Systems EXPECTED KNOWLEDGE Understanding

More information

AC BEHAVIOR OF COMPONENTS

AC BEHAVIOR OF COMPONENTS AC BEHAVIOR OF COMPONENTS AC Behavior of Capacitor Consider a capacitor driven by a sine wave voltage: I(t) 2 1 U(t) ~ C 0-1 -2 0 2 4 6 The current: is shifted by 90 o (sin cos)! 1.0 0.5 0.0-0.5-1.0 0

More information

1. page xviii, line 23:... conventional. Part of the reason for this...

1. page xviii, line 23:... conventional. Part of the reason for this... DSP First ERRATA. These are mostly typos, double words, misspellings, etc. Underline is not used in the book, so I ve used it to denote changes. JMcClellan, February 22, 2002 1. page xviii, line 23:...

More information

EECS 216 Winter 2008 Lab 2: FM Detector Part II: In-Lab & Post-Lab Assignment

EECS 216 Winter 2008 Lab 2: FM Detector Part II: In-Lab & Post-Lab Assignment EECS 216 Winter 2008 Lab 2: Part II: In-Lab & Post-Lab Assignment c Kim Winick 2008 1 Background DIGITAL vs. ANALOG communication. Over the past fifty years, there has been a transition from analog to

More information

Chapter-2 SAMPLING PROCESS

Chapter-2 SAMPLING PROCESS Chapter-2 SAMPLING PROCESS SAMPLING: A message signal may originate from a digital or analog source. If the message signal is analog in nature, then it has to be converted into digital form before it can

More information

Islamic University of Gaza. Faculty of Engineering Electrical Engineering Department Spring-2011

Islamic University of Gaza. Faculty of Engineering Electrical Engineering Department Spring-2011 Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Spring-2011 DSP Laboratory (EELE 4110) Lab#4 Sampling and Quantization OBJECTIVES: When you have completed this assignment,

More information

Digital Signal Processing Lecture 1 - Introduction

Digital Signal Processing Lecture 1 - Introduction Digital Signal Processing - Electrical Engineering and Computer Science University of Tennessee, Knoxville August 20, 2015 Overview 1 2 3 4 Basic building blocks in DSP Frequency analysis Sampling Filtering

More information

ECE 301, final exam of the session of Prof. Chih-Chun Wang Saturday 10:20am 12:20pm, December 20, 2008, STEW 130,

ECE 301, final exam of the session of Prof. Chih-Chun Wang Saturday 10:20am 12:20pm, December 20, 2008, STEW 130, ECE 301, final exam of the session of Prof. Chih-Chun Wang Saturday 10:20am 12:20pm, December 20, 2008, STEW 130, 1. Enter your name, student ID number, e-mail address, and signature in the space provided

More information

Laboratory Assignment 4. Fourier Sound Synthesis

Laboratory Assignment 4. Fourier Sound Synthesis Laboratory Assignment 4 Fourier Sound Synthesis PURPOSE This lab investigates how to use a computer to evaluate the Fourier series for periodic signals and to synthesize audio signals from Fourier series

More information

Introduction to Signals and Systems Lecture #9 - Frequency Response. Guillaume Drion Academic year

Introduction to Signals and Systems Lecture #9 - Frequency Response. Guillaume Drion Academic year Introduction to Signals and Systems Lecture #9 - Frequency Response Guillaume Drion Academic year 2017-2018 1 Transmission of complex exponentials through LTI systems Continuous case: LTI system where

More information

Concordia University. Discrete-Time Signal Processing. Lab Manual (ELEC442) Dr. Wei-Ping Zhu

Concordia University. Discrete-Time Signal Processing. Lab Manual (ELEC442) Dr. Wei-Ping Zhu Concordia University Discrete-Time Signal Processing Lab Manual (ELEC442) Course Instructor: Dr. Wei-Ping Zhu Fall 2012 Lab 1: Linear Constant Coefficient Difference Equations (LCCDE) Objective In this

More information

Lab 8: Frequency Response and Filtering

Lab 8: Frequency Response and Filtering Lab 8: Frequency Response and Filtering Pre-Lab and Warm-Up: You should read at least the Pre-Lab and Warm-up sections of this lab assignment and go over all exercises in the Pre-Lab section before going

More information

Final Exam. EE313 Signals and Systems. Fall 1999, Prof. Brian L. Evans, Unique No

Final Exam. EE313 Signals and Systems. Fall 1999, Prof. Brian L. Evans, Unique No Final Exam EE313 Signals and Systems Fall 1999, Prof. Brian L. Evans, Unique No. 14510 December 11, 1999 The exam is scheduled to last 50 minutes. Open books and open notes. You may refer to your homework

More information

Midterm 1. Total. Name of Student on Your Left: Name of Student on Your Right: EE 20N: Structure and Interpretation of Signals and Systems

Midterm 1. Total. Name of Student on Your Left: Name of Student on Your Right: EE 20N: Structure and Interpretation of Signals and Systems EE 20N: Structure and Interpretation of Signals and Systems Midterm 1 12:40-2:00, February 19 Notes: There are five questions on this midterm. Answer each question part in the space below it, using the

More information

PROBLEM SET 6. Note: This version is preliminary in that it does not yet have instructions for uploading the MATLAB problems.

PROBLEM SET 6. Note: This version is preliminary in that it does not yet have instructions for uploading the MATLAB problems. PROBLEM SET 6 Issued: 2/32/19 Due: 3/1/19 Reading: During the past week we discussed change of discrete-time sampling rate, introducing the techniques of decimation and interpolation, which is covered

More information

GEORGIA INSTITUTE OF TECHNOLOGY. SCHOOL of ELECTRICAL and COMPUTER ENGINEERING. ECE 2026 Summer 2018 Lab #8: Filter Design of FIR Filters

GEORGIA INSTITUTE OF TECHNOLOGY. SCHOOL of ELECTRICAL and COMPUTER ENGINEERING. ECE 2026 Summer 2018 Lab #8: Filter Design of FIR Filters GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2026 Summer 2018 Lab #8: Filter Design of FIR Filters Date: 19. Jul 2018 Pre-Lab: You should read the Pre-Lab section of

More information

Basics of Digital Filtering

Basics of Digital Filtering 4 Basics of Digital Filtering Willis J. Tompkins and Pradeep Tagare In this chapter we introduce the concept of digital filtering and look at the advantages, disadvantages, and differences between analog

More information

Chapter 6 CONTINUOUS-TIME, IMPULSE-MODULATED, AND DISCRETE-TIME SIGNALS. 6.6 Sampling Theorem 6.7 Aliasing 6.8 Interrelations

Chapter 6 CONTINUOUS-TIME, IMPULSE-MODULATED, AND DISCRETE-TIME SIGNALS. 6.6 Sampling Theorem 6.7 Aliasing 6.8 Interrelations Chapter 6 CONTINUOUS-TIME, IMPULSE-MODULATED, AND DISCRETE-TIME SIGNALS 6.6 Sampling Theorem 6.7 Aliasing 6.8 Interrelations Copyright c 2005- Andreas Antoniou Victoria, BC, Canada Email: aantoniou@ieee.org

More information

Infinite Impulse Response (IIR) Filter. Ikhwannul Kholis, ST., MT. Universitas 17 Agustus 1945 Jakarta

Infinite Impulse Response (IIR) Filter. Ikhwannul Kholis, ST., MT. Universitas 17 Agustus 1945 Jakarta Infinite Impulse Response (IIR) Filter Ihwannul Kholis, ST., MT. Universitas 17 Agustus 1945 Jaarta The Outline 8.1 State-of-the-art 8.2 Coefficient Calculation Method for IIR Filter 8.2.1 Pole-Zero Placement

More information

EECS 216 Winter 2008 Lab 2: FM Detector Part I: Intro & Pre-lab Assignment

EECS 216 Winter 2008 Lab 2: FM Detector Part I: Intro & Pre-lab Assignment EECS 216 Winter 2008 Lab 2: Part I: Intro & Pre-lab Assignment c Kim Winick 2008 1 Introduction In the first few weeks of EECS 216, you learned how to determine the response of an LTI system by convolving

More information

ECE 201: Introduction to Signal Analysis

ECE 201: Introduction to Signal Analysis ECE 201: Introduction to Signal Analysis Prof. Paris Last updated: October 9, 2007 Part I Spectrum Representation of Signals Lecture: Sums of Sinusoids (of different frequency) Introduction Sum of Sinusoidal

More information

Signal Processing. Introduction

Signal Processing. Introduction Signal Processing 0 Introduction One of the premiere uses of MATLAB is in the analysis of signal processing and control systems. In this chapter we consider signal processing. The final chapter of the

More information

George Mason University ECE 201: Introduction to Signal Analysis Spring 2017

George Mason University ECE 201: Introduction to Signal Analysis Spring 2017 Assigned: March 7, 017 Due Date: Week of April 10, 017 George Mason University ECE 01: Introduction to Signal Analysis Spring 017 Laboratory Project #7 Due Date Your lab report must be submitted on blackboard

More information

1. In the command window, type "help conv" and press [enter]. Read the information displayed.

1. In the command window, type help conv and press [enter]. Read the information displayed. ECE 317 Experiment 0 The purpose of this experiment is to understand how to represent signals in MATLAB, perform the convolution of signals, and study some simple LTI systems. Please answer all questions

More information

Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot:

Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot: Bode plot From Wikipedia, the free encyclopedia A The Bode plot for a first-order (one-pole) lowpass filter Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and

More information

y(n)= Aa n u(n)+bu(n) b m sin(2πmt)= b 1 sin(2πt)+b 2 sin(4πt)+b 3 sin(6πt)+ m=1 x(t)= x = 2 ( b b b b

y(n)= Aa n u(n)+bu(n) b m sin(2πmt)= b 1 sin(2πt)+b 2 sin(4πt)+b 3 sin(6πt)+ m=1 x(t)= x = 2 ( b b b b Exam 1 February 3, 006 Each subquestion is worth 10 points. 1. Consider a periodic sawtooth waveform x(t) with period T 0 = 1 sec shown below: (c) x(n)= u(n). In this case, show that the output has the

More information

Signals and Systems EE235. Leo Lam

Signals and Systems EE235. Leo Lam Signals and Systems EE235 Leo Lam Today s menu Lab detailed arrangements Homework vacation week From yesterday (Intro: Signals) Intro: Systems More: Describing Common Signals Taking a signal apart Offset

More information

DSP First. Laboratory Exercise #7. Everyday Sinusoidal Signals

DSP First. Laboratory Exercise #7. Everyday Sinusoidal Signals DSP First Laboratory Exercise #7 Everyday Sinusoidal Signals This lab introduces two practical applications where sinusoidal signals are used to transmit information: a touch-tone dialer and amplitude

More information

HW 6 Due: November 3, 10:39 AM (in class)

HW 6 Due: November 3, 10:39 AM (in class) ECS 332: Principles of Communications 2015/1 HW 6 Due: November 3, 10:39 AM (in class) Lecturer: Prapun Suksompong, Ph.D. Instructions (a) ONE part of a question will be graded (5 pt). Of course, you do

More information

PYKC 13 Feb 2017 EA2.3 Electronics 2 Lecture 8-1

PYKC 13 Feb 2017 EA2.3 Electronics 2 Lecture 8-1 In this lecture, I will cover amplitude and phase responses of a system in some details. What I will attempt to do is to explain how would one be able to obtain the frequency response from the transfer

More information

Digital Communication System

Digital Communication System Digital Communication System Purpose: communicate information at certain rate between geographically separated locations reliably (quality) Important point: rate, quality spectral bandwidth requirement

More information

Lecture 9. Lab 16 System Identification (2 nd or 2 sessions) Lab 17 Proportional Control

Lecture 9. Lab 16 System Identification (2 nd or 2 sessions) Lab 17 Proportional Control 246 Lecture 9 Coming week labs: Lab 16 System Identification (2 nd or 2 sessions) Lab 17 Proportional Control Today: Systems topics System identification (ala ME4232) Time domain Frequency domain Proportional

More information

Implementation of Digital Signal Processing: Some Background on GFSK Modulation

Implementation of Digital Signal Processing: Some Background on GFSK Modulation Implementation of Digital Signal Processing: Some Background on GFSK Modulation Sabih H. Gerez University of Twente, Department of Electrical Engineering s.h.gerez@utwente.nl Version 5 (March 9, 2016)

More information

Electrical & Computer Engineering Technology

Electrical & Computer Engineering Technology Electrical & Computer Engineering Technology EET 419C Digital Signal Processing Laboratory Experiments by Masood Ejaz Experiment # 1 Quantization of Analog Signals and Calculation of Quantized noise Objective:

More information

COURSE OUTLINE. Introduction Signals and Noise Filtering: LPF1 Constant-Parameter Low Pass Filters Sensors and associated electronics

COURSE OUTLINE. Introduction Signals and Noise Filtering: LPF1 Constant-Parameter Low Pass Filters Sensors and associated electronics Sensors, Signals and Noise COURSE OUTLINE Introduction Signals and Noise Filtering: LPF Constant-Parameter Low Pass Filters Sensors and associated electronics Signal Recovery, 207/208 LPF- Constant-Parameter

More information

Lecture 7 Frequency Modulation

Lecture 7 Frequency Modulation Lecture 7 Frequency Modulation Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/3/15 1 Time-Frequency Spectrum We have seen that a wide range of interesting waveforms can be synthesized

More information

George Mason University Signals and Systems I Spring 2016

George Mason University Signals and Systems I Spring 2016 George Mason University Signals and Systems I Spring 2016 Laboratory Project #4 Assigned: Week of March 14, 2016 Due Date: Laboratory Section, Week of April 4, 2016 Report Format and Guidelines for Laboratory

More information

Multirate Digital Signal Processing

Multirate Digital Signal Processing Multirate Digital Signal Processing Basic Sampling Rate Alteration Devices Up-sampler - Used to increase the sampling rate by an integer factor Down-sampler - Used to increase the sampling rate by an integer

More information

Chapter 7 Filter Design Techniques. Filter Design Techniques

Chapter 7 Filter Design Techniques. Filter Design Techniques Chapter 7 Filter Design Techniques Page 1 Outline 7.0 Introduction 7.1 Design of Discrete Time IIR Filters 7.2 Design of FIR Filters Page 2 7.0 Introduction Definition of Filter Filter is a system that

More information

Lab S-5: DLTI GUI and Nulling Filters. Please read through the information below prior to attending your lab.

Lab S-5: DLTI GUI and Nulling Filters. Please read through the information below prior to attending your lab. DSP First, 2e Signal Processing First Lab S-5: DLTI GUI and Nulling Filters Pre-Lab: Read the Pre-Lab and do all the exercises in the Pre-Lab section prior to attending lab. Verification: The Exercise

More information

FREQUENCY RESPONSE AND PASSIVE FILTERS LABORATORY

FREQUENCY RESPONSE AND PASSIVE FILTERS LABORATORY FREQUENCY RESPONSE AND PASSIVE FILTERS LABORATORY In this experiment we will analytically determine and measure the frequency response of networks containing resistors, AC source/sources, and energy storage

More information

Objectives. Presentation Outline. Digital Modulation Lecture 03

Objectives. Presentation Outline. Digital Modulation Lecture 03 Digital Modulation Lecture 03 Inter-Symbol Interference Power Spectral Density Richard Harris Objectives To be able to discuss Inter-Symbol Interference (ISI), its causes and possible remedies. To be able

More information

Principles of Communications ECS 332

Principles of Communications ECS 332 Principles of Communications ECS 332 Asst. Prof. Dr. Prapun Suksompong prapun@siit.tu.ac.th 5. Angle Modulation Office Hours: BKD, 6th floor of Sirindhralai building Wednesday 4:3-5:3 Friday 4:3-5:3 Example

More information

IIR Filter Design Chapter Intended Learning Outcomes: (i) Ability to design analog Butterworth filters

IIR Filter Design Chapter Intended Learning Outcomes: (i) Ability to design analog Butterworth filters IIR Filter Design Chapter Intended Learning Outcomes: (i) Ability to design analog Butterworth filters (ii) Ability to design lowpass IIR filters according to predefined specifications based on analog

More information

Communication Engineering Prof. Surendra Prasad Department of Electrical Engineering Indian Institute of Technology, Delhi

Communication Engineering Prof. Surendra Prasad Department of Electrical Engineering Indian Institute of Technology, Delhi Communication Engineering Prof. Surendra Prasad Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 23 The Phase Locked Loop (Contd.) We will now continue our discussion

More information

Lab 6 rev 2.1-kdp Lab 6 Time and frequency domain analysis of LTI systems

Lab 6 rev 2.1-kdp Lab 6 Time and frequency domain analysis of LTI systems Lab 6 Time and frequency domain analysis of LTI systems 1 I. GENERAL DISCUSSION In this lab and the next we will further investigate the connection between time and frequency domain responses. In this

More information

EE 422G - Signals and Systems Laboratory

EE 422G - Signals and Systems Laboratory EE 422G - Signals and Systems Laboratory Lab 3 FIR Filters Written by Kevin D. Donohue Department of Electrical and Computer Engineering University of Kentucky Lexington, KY 40506 September 19, 2015 Objectives:

More information

Instruction Manual for Concept Simulators. Signals and Systems. M. J. Roberts

Instruction Manual for Concept Simulators. Signals and Systems. M. J. Roberts Instruction Manual for Concept Simulators that accompany the book Signals and Systems by M. J. Roberts March 2004 - All Rights Reserved Table of Contents I. Loading and Running the Simulators II. Continuous-Time

More information

II Year (04 Semester) EE6403 Discrete Time Systems and Signal Processing

II Year (04 Semester) EE6403 Discrete Time Systems and Signal Processing Class Subject Code Subject II Year (04 Semester) EE6403 Discrete Time Systems and Signal Processing 1.CONTENT LIST: Introduction to Unit I - Signals and Systems 2. SKILLS ADDRESSED: Listening 3. OBJECTIVE

More information

Digital Communication System

Digital Communication System Digital Communication System Purpose: communicate information at required rate between geographically separated locations reliably (quality) Important point: rate, quality spectral bandwidth, power requirements

More information

Frequency Division Multiplexing Spring 2011 Lecture #14. Sinusoids and LTI Systems. Periodic Sequences. x[n] = x[n + N]

Frequency Division Multiplexing Spring 2011 Lecture #14. Sinusoids and LTI Systems. Periodic Sequences. x[n] = x[n + N] Frequency Division Multiplexing 6.02 Spring 20 Lecture #4 complex exponentials discrete-time Fourier series spectral coefficients band-limited signals To engineer the sharing of a channel through frequency

More information

Design and comparison of butterworth and chebyshev type-1 low pass filter using Matlab

Design and comparison of butterworth and chebyshev type-1 low pass filter using Matlab Research Cell: An International Journal of Engineering Sciences ISSN: 2229-6913 Issue Sept 2011, Vol. 4 423 Design and comparison of butterworth and chebyshev type-1 low pass filter using Matlab Tushar

More information

(Refer Slide Time: 02:00-04:20) (Refer Slide Time: 04:27 09:06)

(Refer Slide Time: 02:00-04:20) (Refer Slide Time: 04:27 09:06) Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 25 Analog Filter Design (Contd.); Transformations This is the 25 th

More information

Continuous-Time Analog Filters

Continuous-Time Analog Filters ENGR 4333/5333: Digital Signal Processing Continuous-Time Analog Filters Chapter 2 Dr. Mohamed Bingabr University of Central Oklahoma Outline Frequency Response of an LTIC System Signal Transmission through

More information

Handout 2: Fourier Transform

Handout 2: Fourier Transform ENGG 2310-B: Principles of Communication Systems Handout 2: Fourier ransform 2018 19 First erm Instructor: Wing-Kin Ma September 3, 2018 Suggested Reading: Chapter 2 of Simon Haykin and Michael Moher,

More information

Subtractive Synthesis. Describing a Filter. Filters. CMPT 468: Subtractive Synthesis

Subtractive Synthesis. Describing a Filter. Filters. CMPT 468: Subtractive Synthesis Subtractive Synthesis CMPT 468: Subtractive Synthesis Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University November, 23 Additive synthesis involves building the sound by

More information

DSP Laboratory (EELE 4110) Lab#10 Finite Impulse Response (FIR) Filters

DSP Laboratory (EELE 4110) Lab#10 Finite Impulse Response (FIR) Filters Islamic University of Gaza OBJECTIVES: Faculty of Engineering Electrical Engineering Department Spring-2011 DSP Laboratory (EELE 4110) Lab#10 Finite Impulse Response (FIR) Filters To demonstrate the concept

More information

Poles and Zeros of H(s), Analog Computers and Active Filters

Poles and Zeros of H(s), Analog Computers and Active Filters Poles and Zeros of H(s), Analog Computers and Active Filters Physics116A, Draft10/28/09 D. Pellett LRC Filter Poles and Zeros Pole structure same for all three functions (two poles) HR has two poles and

More information

ECE5713 : Advanced Digital Communications

ECE5713 : Advanced Digital Communications ECE5713 : Advanced Digital Communications Bandpass Modulation MPSK MASK, OOK MFSK 04-May-15 Advanced Digital Communications, Spring-2015, Week-8 1 In-phase and Quadrature (I&Q) Representation Any bandpass

More information

Lecture 4 Frequency Response of FIR Systems (II)

Lecture 4 Frequency Response of FIR Systems (II) EE3054 Signals and Systems Lecture 4 Frequency Response of FIR Systems (II Yao Wang Polytechnic University Most of the slides included are extracted from lecture presentations prepared by McClellan and

More information

Rotary Motion Servo Plant: SRV02. Rotary Experiment #03: Speed Control. SRV02 Speed Control using QuaRC. Student Manual

Rotary Motion Servo Plant: SRV02. Rotary Experiment #03: Speed Control. SRV02 Speed Control using QuaRC. Student Manual Rotary Motion Servo Plant: SRV02 Rotary Experiment #03: Speed Control SRV02 Speed Control using QuaRC Student Manual Table of Contents 1. INTRODUCTION...1 2. PREREQUISITES...1 3. OVERVIEW OF FILES...2

More information