Chapter 12

The discrete-time Fourier transform

Four lessons in Part V, Discrete-time Fourier analysis. Read them in order, or start anywhere: a prerequisite is a link, never a gate.

Start with 12.1

0 of 4 read4 on the essential pathabout 90 minutes

Ω = 2.00π rad/sample; in (−π, π] it is 0 rad/sample. Wheel: the unit circle with marks n = 0 to 7 at angle Ωn, an arrow from the centre to mark 1 and an arc for Ω; marks 0, 1, 2, 3, 4, 5, 6 and 7 sit on one spot. Stems x[n] = cos(Ωn) for n = 0 to 15, starting 1, 1, 1, 1, 1, 1, 1, 1.

Lesson 1 Essential15 minYou are hereRead

Frequency in discrete time

See digital frequency as a turn per sample, why the frequency axis closes into a circle, and how to convert among its four common scales.

$Plane panel: five unit arrows for n = −2 to 2, nose to tail from the origin, each turned back by Ω times n; the arrow for n = 0 is in the accent colour. Ω = 2.00π rad/sample. The chain ends at the square, X(e^{jΩ}) = 5.00. Strip panel: X(e^{jΩ}) against Ω (rad/sample), from 0 to 2π.$ $Plane panel: five unit arrows for n = −2 to 2, nose to tail from the origin, each turned back by Ω times n; the arrow for n = 0 is in the accent colour. Ω = 2.00π rad/sample. The chain ends at the square, X(e^{jΩ}) = 5.00. Strip panel: X(e^{jΩ}) against Ω (rad/sample), from 0 to 2π.$

Lesson 2 Essential25 minYou are hereRead

The DTFT

Turn each sample back by Ω per sample and add the arrows: a spectrum that repeats every 2π, read as size and angle.

$Size of the spectrum of 0.8ⁿu[n] multiplied by e^{jΩ₁n}, for Ω₁ = 1.00π rad/sample. Solid curve, after the shift: peaks of height 5 at both ends, −π and π, which are the same point. Dashed curve, x alone: a peak of height 5 at 0 rad/sample.$ $Size of the spectrum of 0.8ⁿu[n] multiplied by e^{jΩ₁n}, for Ω₁ = 1.00π rad/sample. Solid curve, after the shift: peaks of height 5 at both ends, −π and π, which are the same point. Dashed curve, x alone: a peak of height 5 at 0 rad/sample.$