Project: Lowpass-to-Bandpass transformation

Design a Chebyshev Type I LPF with the following specifications:

(i) 0.2 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 p₀, p₁, . . ., p_N-1.

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 f_i = 40 kHz and f_u = 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

(a) Your original low pass filter transfer function H_L(S) from HW8 can be expressed as:

H_L(s) = [K/(s/p₀, - 1)(s/p₁ - 1)· · · (s/p_N-1 - 1)].                        (1)

Find the value of the gain K so that H_L(0) has the correct value for your filter.

Background Information on Low Pass to Baird Pass Transformation

The low pass to band pass filter transformation replaces every occurrence of the variable s in your LPF transfer function with s→ω_pQ(s/ω₀ + ω₀/s), where ω_p is the ion pass filter pass band edge frequency in rads/sec, Q is the BPF quality factor defined as Q = ω₀/ω_u-ω_l, ω_u and ω_l are the BPF upper and lower pass band edge frequencies in rads/sec, and ω₀ = √((ω_uω_l) is the BPF's pass band center frequency. Thus, the transfer function H(s) of the new BPF will be

H(s)= H_L (ω_pQ(s/ω₀ + ω⁰/s)).                                        (2)

(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 ω_pQ(s/ω₀ + ω₀/s) - p = 0, find the locations of the corresponding two poles in the BPF, in terms of p, ω_p, ω_u and ω_l.

(c) Using your result from part (b) and your original N poles for your LPF in HW8, list the 2N poles of the BPF, and also plot them on a polar plot using the MATLAB command polar(angle(pn), abs(pn),'X'), where pn is a 2N x 1 array of the poles of your BPF. Please also make a polar plot or the 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 H(s) in equation (2) above, and also the original LPF transfer function H_L(s) in equation (1) above.)

(e) Use MATLAB to plot 20log₁₀|H(f)| for your BPF on the vertical axis (i.e. |H(f)| in dB), and frequency f (in Hz) on the horizontal axis. To do this, just substitute s = j2πf in your expression for H(s) in equation (1), and plot for a range of f sufficient to cover your passband and stopband (suggested range is 37kHz to 63k Hz). In MATLAB, the variable "j" is used for √(-1) 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( [xmin xmax 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, ie, does your plot in part e fail below -50 dB for frequencies at or below 37 kHz, and at or above 63 kHz? If the stop band specs are not met, why do you think that happened?

(g) Plot the phase of H(f) vs. frequency f on another graph. The MATLAB "angle()" function finds the phase of a complex number.

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

(i) extra credit: The LPF-to-BPF transformation s →ω_pQ(s/ω₀ + ω₀/s) is realized at the circuit level by replacing ever., 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).

Note: Credit will not be given if any of MATLAB's built-in Chebysher filter commands or functions are used to make the plots for parts (0 and (g).