A linear-FM chirp signal is a sinusoid whose instantaneous frequency changes linearly with time. The formula for such a signal can be defined by creating a complex exponential signal with a quadratic angle function theta(t). Mathematically, we define theta(t)as theta(t) = 2 pi mu t2 + 2 pi f0t + and the signal is written as x(t) = Re{Ae j theta (t)] = Acos(theta(t)). The derivative of theta(t) yields an instantaneous frequency that changes linearly with time: The linear function of (4) has a slope 2 mu and is equal to f0 at t = 0. Often, we define the \"linear sweep\" by specifying the frequencies at the starting and ending moments, e.g., (starting with) 150Hz at t= 0.5s and (ending with) 1050Hz at t = 5s. From the specification, we obtain the parameters in (2) as and f0 = 50Hz. The signal is thus defined as x (t) = Acos(200 pi-t2 + 100 pi t+(phi). where A and phi are arbitrary. You\'ll need a MATLAB code to generate a linear FM chirp. You may recall make_chirp in Lab03; make a modification to it so that it takes the four parameters as input, beginning time and frequency, and ending time and frequency, plus the sampling rate. Fill in the ?? below to complete the code: Solution Chirp Signal – Frequency Sweeping – FFT and power spectral density Numerous texts are available to explain the basics of Discrete Fourier Transform and its very efficient implementation – Fast Fourier Transform (FFT). Often we are confronted with the need to generate simple, standard signals (sine, cosine, Gaussian pulse, squarewave, isolated rectangular pulse, exponential decay, chirp signal) for simulation purpose. I intend to show (in a series of articles) how these basic signals can be generated in Matlab and how to represent them in frequency domain using FFT. All the signals discussed so far do not change in frequency over time. Obtaining a signal with time-varying frequency is of main focus here. A signal that varies in frequency over time is called “chirp”. The frequency of the chirp signal can vary from low to high frequency (up-chirp) or from high to low frequency (low-chirp). Mathematical Description: A linear chirp signal sweeps the frequency from low to high frequency (or vice-versa) linearly. One approach to generate a chirp signal is to concatenate a series of segments of sine waves each with increasing(or decreasing) frequency in order. This method introduces discontinuities in the chirp signal due to the mismatch in the phases of each such segments. Modifying the equation of a sinusoid to generate a chirp signal is a better approach. The equation for generating a sinusoidal (cosine here) signal with amplitude A, angular frequency 0 and initial phase is x(t)=Acos(0t+)(1) This can be written as a function of instantaneous phase x(t)=Acos((t))(2) where (t)=0+ is the instantaneous phase of the sinusoid and it is linear in time. The time derivative of instantaneous phase (t) is equal to the angular frequency of the sinusoid – which in case is a constant in t.

1. A linear-FM chirp signal is a sinusoid whose instantaneous frequency changes linearly with
time. The formula for such a signal can be defined by creating a complex exponential signal with
a quadratic angle function theta(t). Mathematically, we define theta(t)as theta(t) = 2 pi mu t2 + 2
pi f0t + and the signal is written as x(t) = Re{Ae j theta (t)] = Acos(theta(t)). The derivative of
theta(t) yields an instantaneous frequency that changes linearly with time: The linear function of
(4) has a slope 2 mu and is equal to f0 at t = 0. Often, we define the "linear sweep" by
specifying the frequencies at the starting and ending moments, e.g., (starting with) 150Hz at t=
0.5s and (ending with) 1050Hz at t = 5s. From the specification, we obtain the parameters in (2)
as and f0 = 50Hz. The signal is thus defined as x (t) = Acos(200 pi-t2 + 100 pi t+(phi). where
A and phi are arbitrary. You'll need a MATLAB code to generate a linear FM chirp. You may
recall make_chirp in Lab03; make a modification to it so that it takes the four parameters as
input, beginning time and frequency, and ending time and frequency, plus the sampling rate. Fill
in the ?? below to complete the code:
Solution
Chirp Signal – Frequency Sweeping – FFT and power spectral density
Numerous texts are available to explain the basics of Discrete Fourier Transform and its very
efficient implementation – Fast Fourier Transform (FFT). Often we are confronted with the need
to generate simple, standard signals (sine, cosine, Gaussian pulse, squarewave, isolated
rectangular pulse, exponential decay, chirp signal) for simulation purpose. I intend to show (in a
series of articles) how these basic signals can be generated in Matlab and how to represent them
in frequency domain using FFT.
All the signals discussed so far do not change in frequency over time. Obtaining a signal with
time-varying frequency is of main focus here. A signal that varies in frequency over time is
called “chirp”. The frequency of the chirp signal can vary from low to high frequency (up-chirp)
or from high to low frequency (low-chirp).
Mathematical Description:
A linear chirp signal sweeps the frequency from low to high frequency (or vice-versa) linearly.
One approach to generate a chirp signal is to concatenate a series of segments of sine waves each
with increasing(or decreasing) frequency in order. This method introduces discontinuities in the
chirp signal due to the mismatch in the phases of each such segments. Modifying the equation of
a sinusoid to generate a chirp signal is a better approach.
The equation for generating a sinusoidal (cosine here) signal with amplitude A, angular
frequency 0 and initial phase is
x(t)=Acos(0t+)(1)

2. This can be written as a function of instantaneous phase
x(t)=Acos((t))(2)
where (t)=0+ is the instantaneous phase of the sinusoid and it is linear in time. The time
derivative of instantaneous phase (t) is equal to the angular frequency of the sinusoid – which in
case is a constant in the above equation.
(t)=ddt(t)(3)
Instead of having the phase linear in time, let’s change the phase to quadratic form and thus non-
linear.
(t)=2t2+2f0t+(4)
for some constant . The first derivative the phase, which is the instantaneous angular frequency
becomes
i(t)=ddt(t)=4t+2f0(5)
The time-varying frequency in Hz is given by
fi(t)=2t+f0(6)
In the above equation, the frequency is no longer a constant, rather it is of time-varying nature
with initial frequency given by f0. Thus, from the above equation, given a time duration T, the
rate of change of frequency is given by
k=2=f1f0T(7)
where, f0 is the starting frequency of the sweep, f_1 is the frequency at the end of the duration
T.
The time varying frequency function for a phase derivative equal to i(t)=2(t+f0) is
fi(t)=t+f0(8)
Substituting (6) & (7) in (5)
i(t)=ddt(t)=2(kt+f0)(9)
From (3) and (9)
(t)=i(t)dt=2(kt+f0)dt=2(kt22+f0t)+0=2(kt22+f0t)+0=2(k2t+f0)t+0(10)
where 0 is a constant which will act as the initial phase of the sweep.
Thus the modified equation for generating a chirp signal is given by
x(t)=Acos(2f(t)t+0)(11)
where the time-varying frequency function is given by
f(t)=k2t+f0