| ECE 381: Laboratory 2 |
Winter 2006 |
| Complex exponentials and chirp signals |
January 31 |
|
| Part II. Note on chirp
signals |
MATLAB
notes | Note on chirp signals | Assignment |
|
Chirp signals, also called swept-frequency signals, are sinusoids whose
frequency varies linearly with time. To understand chirp signals, we need the
concept of instantaneous frequency of a sinusoid. Consider the sinusoid
x(t) = cos[Φ(t)] with a general argument
Φ(t). The instantaneous frequency of
x(t), call it
fi(t), is defined as the derivative of
Φ(t) with respect to t, divided by 2π,
i.e., fi(t) = Φ'(t)/2π.
As a simple example, consider the constant-frequency sinusoid, or as we also
say, the single-tone signal x(t) =
cos(2πf0t + θ). The instantaneous frequency of
this signal is found as fi(t) =
f0 for all t, as we intuitively expect.
Let us now try to find the argument of a sinusoid whose instantaneous
frequency varies linearly from f0 at time
t0 to f1 at time
t0+T for some T >
0. That is, we want fi(t0) =
f0 and
fi(t0+T) =
f1. First, set up the equation of the line through
these two points in the time-frequency plane as
fi(t) = f0 +
(f1-f0)(t-t0)/T.
Second, take the running integral of fi(t)
up to time t , and multiply the result by
2π. Thus, we find:
Φ(t) = 2πf0t +
π(f1-f0)(t-t0)2/T
+ θ,
where θ is a phase constant.
From this general analog chirp signal, we can reach its discrete-time
counterpart by way of sampling. Let S be the sampling rate in Hertz (or,
samples per second). Also let ts = 1/S be
the sampling interval in seconds. Now consider the discrete-time signal
x[n] = cos(Φ[n]), where Φ[n] =
Φ(nts), and Φ(t) is as found above
with t0 = 0 and θ = 0 for
simplicity. We can express the resulting general discrete-time argument as
Φ[n] = 2πF0n +
π(F1-F0)n2/TS,
where F0 =
f0/S and F1 =
f1/S are the digital frequencies in cycles per
sample corresponding to the analog frequencies f0
and f1, respectively. Now, assume that
TS is an integer, say N. Then, we reach a general
expression for a discrete-time chirp signal:
x[n] = cos[2πF0n +
π(F1-F0)n2/N],
n = 0, 1, 2, …, N-1.
This signal, which is
defined whether or not there is an underlying analog signal being sampled, is
called an N-sample chirp signal, and its digital frequency varies
linearly from F0 to
F1 in the span of N samples.
We conclude this note by noting that, while analog chirp signals are never
periodic, discrete-time chirp signals can be periodic.
School of Computing and Engineering
University of
Missouri - Kansas City |
Last updated: January 28, 2006
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