6.003 Signal Processing

Week 5, Lecture B:Discrete Fourier Transform (II)

6.003 Fall 2020

Discrete Fourier TransformA new Fourier representation for DT signals:

The DFT has a number of features that make it particular convenient• It is not limited to periodic signals. • It is discrete in both domains, making it computationally feasible

Synthesis equation

Analysis equation

𝑥 𝑛 =

𝑘=0

𝑁−1

𝑋 𝑘 𝑒𝑗2𝜋𝑘𝑁 𝑛

𝑋 𝑘 =1

𝑁

𝑛=0

𝑁−1

𝑥[𝑛] ∙ 𝑒−𝑗2𝜋𝑘𝑁 𝑛

The FFT (Fast Fourier Transform) is an algorithm for computing the DFT efficiently.

𝑥0 𝑛

Two Ways to Think About DFTWe can think about the DFT in two different ways:

1. Think about DFT as Fourier series of N samples of the signal, periodically extended.

We can see why DFT of a single sinusoid is not concentrated in a single k component

𝑋 𝑘 =1

𝑁

𝑛=0

𝑁−1

𝑥[𝑛] ∙ 𝑒−𝑗2𝜋𝑘𝑁 𝑛

DFT

𝑋0 𝑘iDFT

𝑥 𝑛 =

𝑘=0

𝑁−1

𝑋 𝑘 𝑒𝑗2𝜋𝑘𝑁 𝑛

𝑥𝑝 𝑛

Two Ways to Think About DFTWe can think about the DFT in two different ways:

1. Think about DFT as Fourier series of N samples of the signal, periodically extended.

2. Think about DFT as the scaled Fourier transform of a “windowed” version of the original signal.

DFT: Relation to DTFT

𝑋𝑤 Ω =

𝑛=−∞

∞

𝑥𝑤[𝑛] ∙ 𝑒−𝑗Ω𝑛

DTFT

𝑋𝑤 Ω =

𝑛=0

𝑁−1

𝑥𝑤[𝑛] ∙ 𝑒−𝑗Ω𝑛

𝑤[𝑛] = ൜1 0 ≤ 𝑛 < 𝑁0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝑋 𝑘𝑋 𝑘 =

1

𝑁𝑋𝑤(

2𝜋𝑘

𝑁)

Effect of Windowing on Fourier RepresentationsExample: complex exponential signal 𝑥 𝑛 = 𝑒𝑗Ω0𝑛

Compute the DTFT:

𝑥[𝑛] =1

2𝜋න2𝜋

𝑋(Ω) ∙ 𝑒𝑗Ω𝑛 𝑑Ω = 𝑒𝑗Ω0𝑛We need to find 𝑋(Ω) such that :

𝑥 𝑛 =1

2𝜋න−𝜋

𝜋

𝑋 Ω ∙ 𝑒𝑗Ω𝑛 𝑑Ω = 𝑒𝑗Ω0𝑛

𝑋 Ω =

𝑚=−∞

∞

2𝜋𝛿(Ω − Ω0 + 2𝜋𝑚)

Effect of Windowing on Fourier Representations

Apply a rectangular window

𝑥 𝑛 = 𝑒𝑗Ω0𝑛to the complex exponential signal

What the Fourier Transform 𝑋𝑤 Ω look like?

𝑋𝑤 Ω =

𝑛=−∞

∞

𝑥𝑤[𝑛]𝑒−𝑗Ω𝑛

𝑤[𝑛] = ൜1 0 ≤ 𝑛 < 𝑁0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

such that 𝑥𝑤 𝑛 = 𝑥 𝑛 ∙ 𝑤 𝑛 = 𝑒𝑗Ω0𝑛𝑤[𝑛]

=

𝑛=−∞

∞

𝑒𝑗Ω0𝑛𝑤[𝑛]𝑒−𝑗Ω𝑛 =

𝑛=−∞

∞

𝑤[𝑛]𝑒−𝑗(Ω−Ω0)𝑛

= 𝑊(Ω − Ω0)

𝑒𝑗Ω0𝑛𝑤[𝑛]𝐷𝑇𝐹𝑇

𝑊(Ω − Ω0)

Effect of Windowing on Fourier Representations

Let As shown below for N=15

What is the DTFT of 𝑤[𝑛]?

𝑊 Ω =

𝑛=−∞

∞

𝑤[𝑛]𝑒−𝑗Ω𝑛

𝑤[𝑛] = ൜1 0 ≤ 𝑛 < 𝑁0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

=

𝑛=0

𝑁−1

𝑒−𝑗Ω𝑛

𝑊 Ω =1 − 𝑒−𝑗Ω𝑁

1 − 𝑒−𝑗Ω=𝑒−𝑗Ω

𝑁2(𝑒𝑗Ω

𝑁2 − 𝑒−𝑗Ω

𝑁2)

𝑒−𝑗Ω12(𝑒𝑗Ω

12 − 𝑒−𝑗Ω

12)

=sin(Ω

𝑁2)

sin(Ω2)𝑒−𝑗Ω

𝑁−12

If Ω=0, 𝑊 Ω = 0 = 𝑁

If Ω ≠ 0:

From Lecture 04B:

𝑃𝑆(Ω) =sin(Ω(𝑆 +

12)

sin(Ω2)

Effect of Windowing on Fourier Representations

Let As shown below for N=15

What is the DTFT of 𝑤[𝑛]?

𝑊 Ω =

𝑛=−∞

∞

𝑤[𝑛]𝑒−𝑗Ω𝑛

𝑤[𝑛] = ൜1 0 ≤ 𝑛 < 𝑁0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

=

𝑛=0

𝑁−1

𝑒−𝑗Ω𝑛 =

𝑁 Ω = 0

sin(Ω𝑁2)

sin(Ω2)𝑒−𝑗Ω

𝑁−12 , Ω ≠ 0

Effect of Windowing on Fourier Representations

The effect of windowing can be seen from this example of complex exponential signal:

The frequency content of X(Ω) is at discrete frequencies Ω = Ωo + 2πm

The frequency content of Xw(Ω) is most dense at these same frequencies, but is spread out over almost all other frequencies as well.

Effect of windowing: spectrum smear

𝑥𝑤 𝑛 = 𝑒𝑗Ω0𝑛𝑤[𝑛]

𝐷𝑇𝐹𝑇 𝑋𝑤 Ω = 𝑊(Ω − Ω0)

DFT: Relation to DTFT

𝑋 𝑘 =1

𝑁

𝑛=0

𝑁−1

𝑥[𝑛] ∙ 𝑒−𝑗2𝜋𝑘𝑁

𝑛

DFT

𝑋𝑤 Ω =

𝑛=−∞

∞

𝑥𝑤[𝑛] ∙ 𝑒−𝑗Ω𝑛

DTFT

𝑋𝑤 Ω =

𝑛=0

𝑁−1

𝑥𝑤[𝑛] ∙ 𝑒−𝑗Ω𝑛

𝑤[𝑛] = ൜1 0 ≤ 𝑛 < 𝑁0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝑋 𝑘𝑋 𝑘 =

1

𝑁𝑋𝑤(

2𝜋𝑘

𝑁)

Effect of Windowing on Fourier RepresentationsConsidering the example of 𝑥 𝑛 = 𝑒𝑗Ω0𝑛 and 𝑥𝑤 𝑛 = 𝑒

𝑗Ω0𝑛𝑤 𝑛 with Ω0 =2𝜋

15:

One sample is taken at the peak, and the others fall on zeros.

Because the signal is periodic within the analysis window N.

Ω =2𝜋𝑘

𝑁

Effect of Windowing on Fourier RepresentationsConsidering the example of 𝑥 𝑛 = 𝑒𝑗Ω0𝑛 and 𝑥𝑤 𝑛 = 𝑒

𝑗Ω0𝑛𝑤 𝑛 with Ω0 =4𝜋

15:

One sample is taken at the peak, and the others fall on zeros.

Because the signal is periodic within the analysis window N.

Ω =2𝜋𝑘

𝑁

Effect of Windowing on Fourier RepresentationsConsidering the example of 𝑥 𝑛 = 𝑒𝑗Ω0𝑛 and 𝑥𝑤 𝑛 = 𝑒

𝑗Ω0𝑛𝑤 𝑛 with Ω0 =3𝜋

15:

Ω =2𝜋𝑘

𝑁

Now none of the samples fall on zeros.

Effect of Windowing on Fourier RepresentationsConsidering the example of 𝑥 𝑛 = 𝑒𝑗Ω0𝑛 and 𝑥𝑤 𝑛 = 𝑒

𝑗Ω0𝑛𝑤 𝑛 with Ω0 =2.4𝜋

15:

Ω =2𝜋𝑘

𝑁

Generally, the relation between the samples is complicated.

Spectral Blurring & Frequency ResolutionLonger windows provide finer frequency resolution.

The width of the central lobe is inversely related to window length.

𝑊 Ω =sin(Ω

𝑁2)

sin(Ω2)

𝑒−𝑗Ω𝑁−12

Spectral Blurring & Time/Frequency Tradeoff

→ fundamental tradeoff between resolution in frequency and time.

However, longer windows provide less temporal resolution.

𝑤[𝑛] = ൜1 0 ≤ 𝑛 < 𝑁0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Other types of Windows

𝑤2 𝑛 =ൗ𝑁 2 − ൗ

𝑁2 − 𝑛

ൗ𝑁 2

𝑤1[𝑛]

triangular window

Rectangular window

Hann (or “Hanning”) window

Effects of Different Types of WindowsThe DFT coefficients of 𝑥2 𝑛 = cos

3𝜋

64𝑛 analyzed with N = 64:

Triangular window

Hann (or “Hanning”) window:

SummaryThe Discrete Fourier Transform (DFT)

The two different perspectives looking at DFT and insights gained

Synthesis equation

Analysis equation

𝑥 𝑛 =

𝑘=0

𝑁−1

𝑋 𝑘 𝑒𝑗2𝜋𝑘𝑁 𝑛

𝑋 𝑘 =1

𝑁

𝑛=0

𝑁−1

𝑥[𝑛] ∙ 𝑒−𝑗2𝜋𝑘𝑁 𝑛

A central issue in using the DFT is understanding the tradeoff between time and frequency

• Long analysis windows N provide high resolution in frequency but poor resolution in time.

• Short analysis windows N provide high resolution in time but poor resolution in frequency.

The effect of windowing and different types of windows