Project: Lowpass-to-Bandpass transformation

In HENS you were asked to design a Chebys-hev Type I LPF with the following

specifications!

(00.2 dB passband ripple

(ii) Passband edge frequency = 10000 Hz

(iii) Stopband frequency = 13000 Hz

(iv) Stopband attenuation of 50 dB or more

Your completed design specified the filter order N and the poles pa,, pi, PN-E.

The goal of this assignment is to map your LPF design from HW8 into a band pass filter

(BPF) with the following specs!

(i) 0_2 dB pass band ripple

(ii) The lower and upper pass band edge frequencies are 11 = 40 kHz and fu = 60 kHz. Please note that this assignment requires answers to parts (a)-(h) below, with extra credit for part (i), and that you MUST turn in the MATLAB code you used to do this assignment.

(iii) Your original low pass filter transfer function 111(s) from F1W8 can be expressed as:

HL(s)= (s/ pa, —1Xsi p1 — I). — I) (1)

Find the the value of the gain K so that Hai)) has the correct value for your filter_

Background Information on Low Pass to Band Pass Transformation

The low pass to band pass filter transformation replaces every occurrence of the variable

p

s in your LPF transfer function with 4-- , where e is the imipass filter

pass band edge frequency in rads/sec, Q is the BPF quality factor defined as g -

di„

w and col are the BPF upper and lower pass band edge frequencies in rads/sec, and

.0.)0 —.092,11 is the BPFs pass band center frequency. Thus., the transfer function H(s) of the new BPF will be

H (s), HL (op 01 -

s

(b) The LPF-to-BPF transformation maps every pole of the LPF into two poles for the

(

BPF. Let p be a pole of the LPF. By setting mpg —s -1-.:-21 =p 0, find the

tuc, s

locations of the corresponding two poles in the BPF, in terms of p, Culp, ou and caii_

(c) Using your result from part (b) and your original N poles for your LPF in HW8, list

the 2N poles of the BPF, arid. also plot them on a polar plot using the livIATLAB

command polar(angle(pn),abs(pn),X), where pn is a 2N x I array of the poles of

(2)

your BPF. Please also make a polar plot ofthe original N poles of your LPF, and briefly comment on the similarities and differences between the two plots.

(d) Does your BPF have any zeros? If so, how many, and where are they located? (Hint: Carefully examine your BPF transfer function ii(s) in equation (2) above, and also the original LPF transfer function HI(s) in equation ( 1 ) above.)

(e) Use MATLAB to plot 20logiolH(f)1 for your BPF on the vertical axis (Le. !H(f I in dB), and frequency f (in Hz) on the horizontal axis. To do this., just substitute s = j2Trf in your expression for H(s) in equation (1), and plot for a range off sufficient to cover your passband and stopband (suggested range is 37kHz to 63kHz). In MATLAB, the

variable -j- is used for 11 and the -abs(}- function is used to take the magnitude of a complex number. Your plot should show the passband and stopband specifications (as horizontal lines at the appropriate dB values), and demonstrate that your filter meets at least the passband specs. It may be necessary for you to plot a zoomed-in view of the passband, as well as an overall view, in order to demonstrate that your filter meets passband specs. You can do this by using the plot tools to zoom in on your plot, or you can control the upper and lower limits of the x and y axes by giving the command -axis( [xm in xrnax ymin ymax])-, where xmin, xmax (ymin„ ymax) are real numbers specifying the minimum and maximum values on the x (y) axis.

(f) Comment on whether or not your BPF meets the stop band specs, i,e., does your plot in part e fall below —50 dB for frequencies at or below 37 kHz, and at or above 63 kHz? lithe stop band specs are not met, why do you think that happened?

(g) Plot the phase of 11(I) vs. frequency f on another graph, The MATLAB -angle(- function finds the phase of a complex number.

(h) Make sure you turn in a printout of the program you used to compute your poles and plot all your graphs.

(i) 10 points extra credit. The LPF-to-BPF transformation s

realized at the circuit level by replacing every inductor in the original LPF with a two component network, and every capacitor in the original LPF with a different two component network_ Please specify the two component network that replaces the inductor and capacitor. In each case, give: (i) The type and value of the two components; and (ii) how the two components are connected together (hint: they will either be in series or parallel).

— 1:119) ) i

0 3

Note: Credit will not he given if any of MATLABs Chehyshev filter

commands or functions are used to make the plots for parts (e) and (g).

In HENS you were asked to design a Chebys-hev Type I LPF with the following

specifications!

(00.2 dB passband ripple

(ii) Passband edge frequency = 10000 Hz

(iii) Stopband frequency = 13000 Hz

(iv) Stopband attenuation of 50 dB or more

Your completed design specified the filter order N and the poles pa,, pi, PN-E.

The goal of this assignment is to map your LPF design from HW8 into a band pass filter

(BPF) with the following specs!

(i) 0_2 dB pass band ripple

(ii) The lower and upper pass band edge frequencies are 11 = 40 kHz and fu = 60 kHz. Please note that this assignment requires answers to parts (a)-(h) below, with extra credit for part (i), and that you MUST turn in the MATLAB code you used to do this assignment.

(iii) Your original low pass filter transfer function 111(s) from F1W8 can be expressed as:

HL(s)= (s/ pa, —1Xsi p1 — I). — I) (1)

Find the the value of the gain K so that Hai)) has the correct value for your filter_

Background Information on Low Pass to Band Pass Transformation

The low pass to band pass filter transformation replaces every occurrence of the variable

p

s in your LPF transfer function with 4-- , where e is the imipass filter

pass band edge frequency in rads/sec, Q is the BPF quality factor defined as g -

di„

w and col are the BPF upper and lower pass band edge frequencies in rads/sec, and

.0.)0 —.092,11 is the BPFs pass band center frequency. Thus., the transfer function H(s) of the new BPF will be

H (s), HL (op 01 -

s

(b) The LPF-to-BPF transformation maps every pole of the LPF into two poles for the

(

BPF. Let p be a pole of the LPF. By setting mpg —s -1-.:-21 =p 0, find the

tuc, s

locations of the corresponding two poles in the BPF, in terms of p, Culp, ou and caii_

(c) Using your result from part (b) and your original N poles for your LPF in HW8, list

the 2N poles of the BPF, arid. also plot them on a polar plot using the livIATLAB

command polar(angle(pn),abs(pn),X), where pn is a 2N x I array of the poles of

(2)

your BPF. Please also make a polar plot ofthe original N poles of your LPF, and briefly comment on the similarities and differences between the two plots.

(d) Does your BPF have any zeros? If so, how many, and where are they located? (Hint: Carefully examine your BPF transfer function ii(s) in equation (2) above, and also the original LPF transfer function HI(s) in equation ( 1 ) above.)

(e) Use MATLAB to plot 20logiolH(f)1 for your BPF on the vertical axis (Le. !H(f I in dB), and frequency f (in Hz) on the horizontal axis. To do this., just substitute s = j2Trf in your expression for H(s) in equation (1), and plot for a range off sufficient to cover your passband and stopband (suggested range is 37kHz to 63kHz). In MATLAB, the

variable -j- is used for 11 and the -abs(}- function is used to take the magnitude of a complex number. Your plot should show the passband and stopband specifications (as horizontal lines at the appropriate dB values), and demonstrate that your filter meets at least the passband specs. It may be necessary for you to plot a zoomed-in view of the passband, as well as an overall view, in order to demonstrate that your filter meets passband specs. You can do this by using the plot tools to zoom in on your plot, or you can control the upper and lower limits of the x and y axes by giving the command -axis( [xm in xrnax ymin ymax])-, where xmin, xmax (ymin„ ymax) are real numbers specifying the minimum and maximum values on the x (y) axis.

(f) Comment on whether or not your BPF meets the stop band specs, i,e., does your plot in part e fall below —50 dB for frequencies at or below 37 kHz, and at or above 63 kHz? lithe stop band specs are not met, why do you think that happened?

(g) Plot the phase of 11(I) vs. frequency f on another graph, The MATLAB -angle(- function finds the phase of a complex number.

(h) Make sure you turn in a printout of the program you used to compute your poles and plot all your graphs.

(i) 10 points extra credit. The LPF-to-BPF transformation s

realized at the circuit level by replacing every inductor in the original LPF with a two component network, and every capacitor in the original LPF with a different two component network_ Please specify the two component network that replaces the inductor and capacitor. In each case, give: (i) The type and value of the two components; and (ii) how the two components are connected together (hint: they will either be in series or parallel).

— 1:119) ) i

0 3

Note: Credit will not he given if any of MATLABs Chehyshev filter

commands or functions are used to make the plots for parts (e) and (g).

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