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Lecture 4: Frequency domain representation, DTFT, IDTFT, DFT, IDFTPowerPoint Presentation

Lecture 4: Frequency domain representation, DTFT, IDTFT, DFT, IDFT

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### Lecture 4: Frequency domain representation, DTFT, IDTFT, DFT, IDFT

### Some history

### Frequency domain representation

### Frequency domain representation (cont)

### Frequency domain representation (cont 2)

### Frequency domain representation (cont 3)

### Frequency domain representation (cont 5)

### Frequency domain representation (cont 6)

### DTFT

### DTFT (cont)

### IDTFT

### Back to ideal filters

### DTFT properties

### DTFT examples (cont)

### DTFT examples (cont 2)

### How to measure frequency response of an actual (unknown) filter?

### DFT and IDFT filter?

### DFT and IDFT (cont) filter?

### FFT filter?

### Relation between DTFT and DFT filter?

### Relation between DTFT and DFT (cont) filter?

### Relation between DTFT and DFT (cont 2) filter?

### DFT properties filter?

### DFT properties (cont) filter?

### DFT properties (cont 2) filter?

### Linear filtering via DFT filter?

### Linear filtering via DFT (cont) filter?

### Linear filtering via DFT (cont 2) filter?

### N- filter?point DFTs of 2 real sequences via a single N-point DFT

### Summary filter?

Instructor: Dr. Gleb V. Tcheslavski

Contact:[email protected]

Office Hours: Room 2030

Class web site:http://ee.lamar.edu/gleb/dsp/index.htm

Jean Baptiste Joseph Fourier was born in France in 1768. He attended the Ecole Royale Militaire and in 1790 became a teacher there. Fourier continued his studies at the Ecole Normale in Paris, having as his teachers Lagrange, Laplace, and Monge. Later on, he, together with Monge and Malus, joined Napoleon as scientific advisors to his expedition to Egypt where Fourier established the Cairo Institute.

In 1822 Fourier has published his most famous work: The Analytical Theory of Heat. Fourier showed how the conduction of heat in solid bodies may be analyzed in terms of infinite mathematical series now called by his name, the Fourier series.

(4.3.1)

frequency

(4.3.2)

complex exponent of 0

complex exponent of -0

A sinusoidal signal is represented by TWO complex exponents of opposite frequencies in the frequency domain.

(4.4.1)

(4.4.2)

Iff it exists!

(4.4.3)

(4.4.4)

has a period of 2

For an arbitrary real LTI system:

Symmetric with respect to

Anti-symmetric with respect to

Combining(4.3.2)and(4.4.4) – back to our sinusoid!

(4.6.1)

(4.6.2)

(4.6.3)

(4.6.4)

LTI filtering:

from the input

due to the input

same as the input

change due to the system

phase change due to the system

Via design, we manipulate H(ej), therefore, hn, and, finally, manipulate the coefficients in the Linear Constant Coefficient Difference Equation (LCCDE)

for an LTI:

(4.8.1)

We don’t need systems of order higher than 2: can always make cascades.

for a real, LTI, BIBO system:

(4.8.2)

effects of filtering

We cannot observe ANY frequency components in the output that are not present in the input (in steady state). We may see less when

In continuous time:

(4.9.1)

signal

noise

const

delay

(4.9.2)

We need a constant magnitude and linear phase for the frequencies of interest.

LPF

HPF

BPF

BSF

Ideal filters:

Ideal filters are non-realizable!

(4.11.1)

if exists

(4.11.2)

(4.11.3)

What’s about convergence???

1. Absolute convergence:

(4.11.4)

(4.11.5)

Absolutely summable sequences always have finite energy. However, finite energy sequences are not necessary absolutely summable.

must be

2. Mean-square convergence:

(4.12.1)

The total energy of the error must approach zero, not an error itself!

(4.13.1)

IDTFT:

(4.13.2)

Combining (4.11.1) and (4.12.2)

(4.13.3)

(4.13.4)

shows where xn “lives” in the frequency domain.

(4.14.1)

Ideal LPF:

1

Using IDTFT:

0

c

2

(4.14.2)

- The response in (4.14.2) is not absolutely summable, therefore, the filter is not BIBO stable!
- The response in (4.14.2) is not causal and is of an infinite length.

As a result, the filter in (4.14.1) is not realizable.

Similar derivations show that none of the ideal filters in slide 9 is realizable.

(4.15.1)

(4.15.2)

(4.15.3)

(4.15.4)

(4.15.5)

(4.15.6)

continuous, periodic functions

(4.15.6)

We can re-work the Parseval’s theorem (4.15.6) as follows:

(4.18.1)

energy density (spectrum)

Autocorrelation function:

(4.18.2)

One obvious problem with DTFT is that we can never compute it since xn needs to be known everywhere! which is impossible!

Therefore, DTFT is not practical to compute.

Often, a finite dimension LTI system is described by LCCDE:

(4.19.1)

practical (finite dimensions)

(4.19.2)

Prediction of steady-state behavior of LCCDE

1. Perform two I/O experiments:

(4.20.1)

(4.20.2)

2. Analyze these measurements and form:

(4.20.3)

(4.20.4)

(4.20.5)

That’s a good way to measure/estimate a frequency response for every .

Consider an N-sequence xn (at most N non-zero values for 0 n N-1)

(4.21.1)

uniformly spaced frequency samples

DFT:

(4.21.2)

Finite sum! Therefore, it’s computable.

(4.21.3)

(4.21.1) can be rewritten as:

(4.21.4)

(4.21.5)

Btw, DFT is a sampled version of DTFT.

Let us verify (4.21.5). We multiply both sides by

(4.22.1)

(4.22.2)

(4.22.3)

(4.22.4)

(4.23.1)

In the matrix form:

(4.23.2)

where:

(4.23.3)

(4.23.4)

(4.23.5)

(4.23.6)

(4.23.7)

This is actually FFT…

1. Sampling of DTFT

(4.24.1)

(4.24.2)

(4.24.3)

yn is an infinite sum of shifted replicas of xn. Iff xn is a length M sequence (M N) than yn = xn. Otherwise, time-domain aliasing xncannot be recovered!

2. DTFT from DFT by Interpolation

Let xn be a length N sequence:

Let us try to recover DTFT from DFT (its sampled version).

(4.25.1)

(4.25.2)

(4.25.3)

It’s possible to determine DTFT X(ej) from its uniformly sampled version uniquely!

3. Numerical computation of DTFT from DFT

Let xn is a length N sequence:

defined by N uniformly spaced samples

We wish to evaluate at more dense frequency scale.

(4.26.1)

Define:

zero-padding

(4.26.2)

(4.26.3)

No change in information, no change in DTFT… just a better “plot resolution”.

1. Circular shift

xn is a length N sequence defined for n = 0,1,…N-1.

An arbitrary shift applied to xn will knock it out of the 0…N-1 range.

Therefore, a circular shift that always keeps the shifted sequence in the range 0…N-1 is defined using a modulo operation:

(4.28.1)

(4.28.2)

2. Circular convolution

A linear convolution for two length N sequences xn and gn has a length 2N-1:

(4.29.1)

A circular convolution is a length-N sequence defined as:

(4.29.2)

N

N

(4.29.3)

Procedure: take two sequences of the same length (zero-pad if needed), DFT of them, multiply, IDFT: a circular convolution.

Example:

(4.30.1)

Take N frequency samples of (4.30.1) and then IDFT:

(4.30.2)

aliased version of xn

The results of circular convolution differ from the linear convolution “on the edges” – caused by aliasing.

To avoid aliasing, we need to use zero-padding…

Often, we need to process long data sequences; therefore, the input must be segmented to fixed-size blocks prior LTI filtering. Successive blocks are processed one at a time and the output blocks are fitted together…

We can do it by FFT: IFFT{FFT{x}FFT{h}}…

Problem: DFT implies circular convolution – aliasing!

(4.31.1)

Assuming that hn is an M-sequence, we form an N-sequence (L - block length):

(4.31.2)

N >> M; L >> M; N = L + M - 1 and is a power of 2

Next, we compute N-point DFTs of xm,n and hn, and form

(4.32.1)

- no aliasing!

Since each data block was terminated with M -1 zeros, the last M -1 samples from each block must be overlapped and added to first M – 1 samples of the succeeded block.

An Overlap-Add method.

Alternatively:

Each input data block contains M -1 samples from the previous block followed by L new data samples; multiply the N-DFT of the filter’s impulse response and the N-DFT of the input block, take IDFT.

Keep only the last L data samples from each output block.

The first block is padded by M-1 zeros.

An Overlap-Save method.

Let gn and hn are two length N real sequences.

Form xn = gn + jhn Xk

(4.37.1)

(4.37.2)

(4.37.3)

(4.37.4)

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