Could someone please complete this homework assignment?

MRD101 – ASSIGNMENT #1 NAME: _________________________

1. A signal is described by the graph pictured below. Assume that the mathematical description of the signal is

that of a sine function.

Answer the following questions:

voltage (mV)

time (µs)

a) What is the period of the signal in µs? Show clearly on the graph how you determined this value.

b) Determine the linear frequency f of the signal. Express your answer in kHz.

c) Determine the angular frequency ω of the signal. Write the answer in scientific notation.

d) Determine the time shift of the signal in µs. Show clearly on the graph how you determined this value.

e) Write the mathematical expression that describes the signal.

2. The magnetic field as a function of distance is given by B = 0.050 cos(2.5π (x – 0.1)) where x is measured in
metres and the magnetic field B is in Tesla (T).

a) What is the wavenumber associated with the function in cycles/m?

b) Sketch two complete wavelengths of the function on the graph below:

3. In an MRI acquisition, the recovery of the longitudinal magnetization is described by the function:

( ) ⎟
⎠

⎞
⎜
⎝

⎛
−=

−
11 T

t

oz eMtM where T1 is a time constant which depends on type of tissue present.

The T1 values for two particular tissues are:

Tissue #1 T1 = 1500 ms = 1.500 s
Tissue #2 T1 = 500 ms = 0.500 s

a) Plot a graph of the two magnetization recovery curves on the blank grid on the following page. Use time
values that are multiples of the respective time constants for each tissue t = T1, 2T1, 3T1,,,, (There is no
need to plot beyond five time constants because by this time the signal has recovered to within 1% of its
maximum value). Try to draw the curves as neatly as possible.

0.2Mo

0.4Mo

0.6Mo

0.8Mo

1.0Mo

b) Which signal recovers faster to its maximum value? How is this recovery related to the value of the time
constant T1?

c) Estimate the time at which the signals have their maximum difference in strength.

4. In an MRI acquisition, the decay of the transverse magnetization is described by the function:

( ) 2T
t

oeMtM
−

⊥ = where T2 is a time constant which depends on type of tissue present.

For grey matter, T2 = 90 ms. Assume that initially Mo = 100.

a) What is the value of the transverse magnetization after a time t = 180 ms?

b) What is the value of the transverse magnetization after a time t = 300 ms?

5. Consider the sinc function presented below. Write the mathematical expression representing the graph.

6. Convert the complex number c i0.50.7 += to polar form.

• MRD 101 – ASSIGNMENT #1 NAME: _________________________