SME
Signals & Systems · Problem type
Spectrum of a rectangular pulse
Use this for “A 2 V pulse lasts 1 ms. Find its spectrum at 0 Hz and at 500 Hz, and the first frequency where it is zero.”
Chart
X(500) ≈ 1.273
This is an example. Type your own values to start a new problem.
X(0) (mV·s)
=
A (V)
×
τ (ms)
f₁ (Hz)
=
1
÷
τ (ms)
X(f) (mV·s)
=
X(0) (mV·s)
×
sinc(
f (Hz)
×
τ (ms)
)
Formulas
X(0)
=
A
×
τ
2 = 2 × 1
f₁
=
1
τ
1
=
1
1
X(
f
)
=
X(0)
×
sinc(
f
×
τ
)
1.273 = 2 × sinc(0.5 × 1)
Formulas are worked in kHz. The step-by-step shows the conversions.
Assumptions
•
The pulse is A high for τ, centred on t = 0, so X(f) = Aτ sinc(fτ) is real.
•
sinc(x) = sin(πx) ÷ (πx), with sinc(0) = 1: the normalized sinc of signals texts.
•
With τ in ms, Aτ is in V·ms (the same as mV·s) and 1 ÷ τ is in kHz.
Step-by-step
WE KNOW
A = 2 V, τ = 1 ms, f = 500 Hz
FIND
spectrum at 0 Hz (area) (X(0)), first zero of the spectrum (f₁), spectrum at f (X(f))
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to kHz).
f = 500 Hz = 0.5 kHz (1 kHz = 1000 Hz)
Step 2 · Find spectrum at 0 Hz (area): X(0)
Use X(0) = A × τ
At 0 Hz the transform is the pulse’s area: its height times its width.
X(0) = 2 × 1
X(0) = 2 mV·s
Step 3 · Find first zero of the spectrum (f₁)
Use f₁ = 1 ÷ τ
sinc(fτ)
is
first
0
at
fτ
=
1,
so
the
spectrum’s
first
zero
is
at
1
τ
.
f₁
=
1
1
f₁
=
1
kHz
Step 4 · Find spectrum at f (X(f))
Use X(f) = X(0) × sinc(f × τ)
The
area
at
0
Hz,
scaled
by
sinc(fτ):
the
spectrum
ripples
down
through
zeros
at
whole
multiples
of
1
τ
.
X(
f
)
=
2
×
sinc(0.5
×
1)
X(
f
)
=
2
×
0.6366
X(
f
)
=
1.273
mV·s
Step 5 · Convert the answers
Change each answer back to the unit you chose.
f₁ = 1 kHz = 1,000 Hz (1 kHz = 1000 Hz)
Check
Put the numbers back in. Both sides should match.
2 = 2 × 1 ✓
1
=
1
1
✓
1.273 = 2 × sinc(0.5 × 1) ✓
Topic check
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