The goal of this assignment is to train students on the concepts related to the generalization of the HB technique from the single-tone to the multi-tone excitation. The target

circuit used for this purpose is the doubly-balanced mixer circuit shown in Figure 1.

VDD VDD

RD RD

RL

VRF1 VRF2

VLO1 VLO2

Vout1 Vout2

Xb

Xa Xb

Xa Xb

Xb

Xa

Xa

IO

Figure 1: The circuit schematic

This circuit is simulated by two sources of two different frequencies. The first source

is the output of some local oscillator, whose waveform is given by

vLO1(t) = −vLO2(t) = 0.125 × sin

2 × π × 109 × t

(1)

and the second source is the waveform of some RF frequency given by

vRF1(t) = −vRF2(t) = 0.0125 × sin

2 × π × 900 × 106 × t

(2)

The output of the circuit is a differential output taken between the nodes vout1 −vout2. Thus,

the input sources to this circuit is two-tone ω1 = 2π × 1GHz and ω2 = 2π × 0.9GHz, and

hence its response will contain frequencies at multiples, and combinations of multiples, of

ω1 and ω2. The circuit is designed to filter out all the components at the output nodes except

for the component at the difference between the frequencies, that is, the output vout1 − vout2

will have one dominant frequency ω1 − ω2 = 2π × 100MHz.

The main objective in this assignment is to perform the diamond truncation method

and compute the Fourier coefficients of the truncated frequency spectrum using the HB

technique based on the artificial frequency mapping technique. This objective is broken

down into the following several parts.

Part (a)

In this part of the assignment, you are required to develop a MATLAB program to perform the diamond truncation for the two-tone ω1 and ω2. This should be a one file, called

compute diamond trunc.m, which will take as input the two tones and f1, f2 in Hz

and the diamond truncation index, K and will produce as output all the frequencies grouped

under diamond truncation scheme. Thus, the first line in this program should look like the

following line,

1 function freq = compute_diamond_trunc(f_1,f_2,K)

where,

• f_1 and f_2 are the frequencies of the two tones in Hz,

• K is the truncation bound used by the diamond truncation.

• freq is the a vector of all frequencies, in Hz, extracted from the diamond truncation,

including DC.

and the program file should be called obviously compute_diamond_trunc.

Part (b)

In this part, your task will be to compute the Fourier coefficients of the source vector, b(t),

for the mixer circuit in Figure 1, and use them to construct the vector B¯ of the HB balance

problem. The Fourier coefficients of b(t), are denoted by B0 ∈ R

N , BC

(k1,k2) ∈ R

N and

BS

(k1,k2) ∈ R

N . For a diamond truncation with a truncation index K, only elements that

satisfy |k1| + |k2| ≤ K, need be included in the truncated set of indices ΛM, as shown by

ΛM =

(k1, k2) ∈ Z

2

: |k1| ≤ K1, |k1| + |k2| < K, k2 6= 0 if k1 < 0

\ (k1 = k2 = 0)

IMPORTANT NOTE:

The order you choose to stack the vectors B0, BC

(k1,k2)

and BS

(k1,k2)

in the vector

B¯ should be the same order you use in ordering the frequency components in the function developed in the previous part, freq = compute_diamond_trunc(f_1,f_2,K).

The MATLAB function that you will develop in this part should be general enough to

accomodate any truncation index K used in the diamond truncation scheme. Inside this

function, you are free to assume and use prior knowledge specific to the Mixer circuit

2

above, such as the values of the frequencies of the sources used in the circuit, the indices

of the sources in the MNA time-domain vector b(t), the values of the cos vs the sin component, etc.

The calling syntax of the function developed should mirror exactly the following syntax

1 [freq B_bar] = compute_mt_source_coeffs(K)

where

• freq is a vector of frequencies in the truncated spectrum collected by ΛM in addition

to the DC at the top. This should be the same vector of frequencies returned by the

function of part (a) and in the same order used above.

• B_bar This output argument should hold the vector B¯ in the HB equations, ordered

in harmonic-major mode.

Hints The previous assignment should have given you enough information on how to

construct the B¯ vector in a very limited situation, in the sense that you did not consider the

truncation index (or indices). Your task in this assignment is to make the construction of this

vector automatically linked to a general choice of the truncation index K within a diamond

truncation and the sequence that you choose for ordering the elements of the truncated

double indices in ΛM. You must create the entire B¯ vector based on the truncation index

K in the diamond truncation. Notice that the order you select to sequence the frequency

components is impacted by K and that, in turn, impacts the vector B¯.

The corrector of the assignment will run your function for different values of K, and

will compare the B_bar returned by your function against the correct B¯ vector.

Part (c)

In this part you are required to develop a MATLAB function that computes the permutation

matrix PAFM used in the technique of Artificial Frequency Mapping (AFM). PAFM should

permute the Fourier coefficients for a set of two-tones frequencies truncated by a diamond

truncation scheme with truncation index K into a set of Fourier coefficients for frequencies

that are multiple of an artificial frequency λ0.

The calling syntax should follow the syntax shown below

1 P_AFM = compute_permute_AFM(K)

The input to this function K is the truncation index K to be used in the diamond truncation scheme. The output P_AFM should be the permutation matrix PAFM.

IMPROTANT to NOTE.

PAFM ultimately depends on the sequence that you choose to order the diamond-truncated

set. You MUST adopt the same order that you used in developing the function in part (a),

freq = compute_diamond_trunc(f_1,f_2,K).

3

Hint The previous assignment requested that you construct PAFM for specific values

of truncation indices, be it box- or diamond-truncation. This assignment requests that you

make the construction of such a matrix automatically adjusted for a general index K in a

diamond truncation.

Part (d)

In this part you are asked to develop and test the multi-tone HB technique for the circuit

shown in Fig. 1. The Matlab program expected from you is a function, which takes the

following form

1 [freq, X0, Xc, Xs] = compute_mthb_resp(netlist_file,K,node_name)

where netlist_file should provide the path of the file describing the mixer circuit,

e.g., ../netlists/doubly_balanced_mixer.sp, while the input argument K is

similar to the input used in the previous parts, i.e. it defines the bounds of the diamond truncation scheme, and node name is a string argument that defines the name of the node that

we wish to compute its Fourier spectrum. Most likely, this will be nvout1 or nvout2.

The output of the above function are as follows

• freq is a vector of the truncated frequency spectrum obtained from the diamond

truncation (the original not the artificial one) excluding the DC (zero) frequency.

• X0 is a scalar value representing the DC component of the spectrum of the node

node name provided as the input argument,

• Xc and Xs are vector whose sizes are both identical to the size of the vector freq,

and their numerical values are the values of the Fourier coefficients (respectively,

cosines and sines) of the spectrum of the node node name provided as the input

argument.

Hints

• Use the code that you developed in the previous three parts (a), (b) and (c) to develop

the MATLAB function required in this part (d).

• The code in parts (a), (b) and (c) does not need you to use HiSPCE directly. So it can

be developed and invoked independently without the need to call HiSPICE-Matlab

interface functions, except for the function in part (b) which requires the index of

entries of the sources in the circuit to enable computing the vector B¯. However, you

can find out the indices of those entries offline using HiSPICE, or you may use prior

knowledge of those sources that you utilized in the previous assignment, assignment

4.

4

Part (e)

This part of this assignment requires you to develop a function to plot the time-domain

response, x(t), of the output node (e.g. nvout1) constructed through using the Fourier

coefficients computed from the previous part. By visualizing the steady-state response

in the time-domain you can readily verify and demonstrate that your code is functioning

correctly and without any operation or conceptual bugs.

The syntax of the function that you to develop in this part should mirror the syntax

below

1 [x_time, time_points]=compute_td_from_mthb(freq,X0,XC,XS,n_time_points)

The input arguments to this function are as follows

• freq,X0,Xs,Xs are the outputs returned from the function compute_mthb_resp

developed in part (d).

• n_time_points is an integer number defining the number of time samples at

which x(t) is to be computed.

The output arguments of this function are as follows

• x_time is a column vector whose length is equal to the input argument n_time_points.

This vector should hold the values of x(t0), x(t1), x(t2), · · · x(tP ), where

P = n_time_points − 1.

• time_pointsThis is a column vector whose length is equal to the input argument n_time_points. This vector should hold the values of the sampling instants

t0, t1, t2, · · · , tP , where

P = n_time_points − 1.

.

Hints

• In the above, use t0 equal to 0.

• In order to develop this function correctly, it is very important to know and undertand what is the common or global period T0 for a waveform composed multi-tone

periodical waveforms, i.e., quasi-periodic waveforms. Specifically, the common period T0 will define the time range on which the waveform x(t) is to be sampled at

the required number of sampling points. You will need to read the notes carefully to

figure out the common period in this program.

5

Part (f)

The final part of the assignment requires you to summarize the results of running your

programs in a PDF format. The PDF file should demonstrate the following,

(a) Analysis showing the value of, and how you computed, the common or global period

for the waveforms in the mixer circuit.

(b) Graphs showing the time-domain results obtained from your programs for the voltages at nodes nVout1,nVout2,n4,n5 in the circuit.

(c) A graph showing the result obtained at the differential output voltage, that is

v(nVout1) - v(nVout2).

(d) A graph showing the comparison between some of the above results and corresponding results obtained from the commercial simulator HSPICE, which will be released

in due time.

IMPORTANT INSTRUCTIONS. What you need to submit

You are required to submit only the following files (DO NOT compress them in one

zipped file), and you must submit them in the following order,

(a) MATLAB file compute_diamond_trunc.m

(b) MATLAB file compute_mt_source_coeffs.m

(c) MATLAB file compute_permute_AFM.m

(d) MATLAB file compute_mthb_resp.m. This last one must include all of the

above functions defined internally. The assignments corrector will invoke this function and examine its output.

(e) MATLAB file test_bench.m.

This file is only used to call any or all of the above functions. It is NOT a place where

any function should be defined. A simplified way to represent this file may be shown

below:

6

1 % This is a text bench file used to launch the other

2 % programs developed by

3 % Student Name : Emad Gad

4 % Student Number :12345678

5 clear all

6 close all

7 init_globals

9 netlist_file_name = ’../netlists/doubly_balance--...-.sp’

10 K = 5;

11 f_1 = 1e9;

12 f_2 = 0.1e9;

13 node_name = ’nVout1’;

14 n_time_points = 10000;

15 freq = compute_diamond_trunc(f_1,f_2,K);

16 [freq B_bar] = compute_mt_source_coeffs(K);

17 P_AFM = compute_permute_AFM(K);

18 [freq, X0, Xc, Xs] = ...

19 compute_mthb_resp(netlist_file_name,K,node_name);

20 [x_time, time_points]=...

21 compute_td_from_mthb(freq,X0,Xc,Xs,n_time_points);

22 plot(time_points,x_time)

24 % End of file: no more code to go after this points

(f) A PDF file for your report containing all the items described by part (f).

The PDF file should be named using your student ID and last name, e.g. 1234567 Gad.pdf

DO NOT

• DO NOT compress your files or submit a single .zip or .rar archive file.

• NEVER use clear all inside a function, since it wipes all input arguments.

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