formulário de sistemas e sinais - …¡rio de sistemas e sinais (leic – alameda – 2007/2008 –...

12
(Agrafar o formulário !) Formulário de Sistemas e Sinais (LEIC – Alameda – 2007/2008 – 2º Semestre) Nome do Aluno: ________________________________________________________ Número do Aluno: ______________________________________________________ 1

Upload: vanmien

Post on 21-May-2018

213 views

Category:

Documents


0 download

TRANSCRIPT

(Agrafar o formulário !)

Formulário

de

Sistemas e Sinais

(LEIC – Alameda – 2007/2008 – 2º Semestre)

Nome do Aluno: ________________________________________________________ Número do Aluno: ______________________________________________________

1

Aperiodic timeContinuous frequency

Periodic timeDiscrete frequency

Aperiodicfrequency

Continuoustime

CTFT� ContSignals� ContSignals

���� �

��

��

�����������

InverseCTFT� ContSignals� ContSignals

���� ��

��

��

��

����������

FourierSeries�� ContPeriodic� � DiscSignals

�� ��

��

�������������

InverseFourierSeries��DiscSignals� ContPeriodic�

���� �

������

��������

Periodicfrequency

Discretetime

DTFT� DiscSignals� ContPeriodic��

���� �

������

���������

InverseDTFT� ContPeriodic�� � DiscSignals

���� ��

��

��

��

����������

DFT�� DiscPeriodic� � DiscPeriodic�

�� �

�������

�����������

InverseDFT�� DiscPeriodic� � DiscPeriodic�

���� ��

�������

��������

The DFS and the DFT

The discrete-time Fourier series (DFS) expansion for � � DiscPeriodic� is

� � � Integers� ���� �����

���

���������

where �� � ���� (radians/sample). The Fourier series coefficients can be found using the formula

� � Integers� �� ��

����

���

���������� �

For historical reasons, the discrete Fourier transform (DFT) is the DFS with slightly differentscaling. It is defined by

� � � Integers� ���� � �

�������

� �

��������

� � Integers� ��

� ��������

���������� �

Obviously, the DFT coefficients are related to the DFS coefficients by

� �

� � ����2

Signal DFT

� � � Integers�

���� � ���� ��

where � �� �.

� � � Integers�

� �� �

�� if � � �� otherwise

� � � Integers�

���� � ���������

where � �� �.

� � � Integers�

� �� �

���� if � � ���� if � � � otherwise

� � � Integers�

���� � ����������

where � �� �.

� � � Integers�

� �� �

����� if � � ������ if � � � otherwise

� � � Integers�

���� � �

� � � Integers�

� �� �

�� if � � �� otherwise

Square wave:

� � � �� �� � � � � �� ���

���� �

�� if � � or � � ���� otherwise

� � � Integers�

� �� �

������ � � �����

���������

Impulse train:

� � � Integers�

���� �

���

��� ���

� � � Integers�

� �� � �

Discrete Fourier transform of periodic signals. The fundamentalfrequency is �� � ���(, where ( is the period. For the complex exponentialand sinusoidal signals, the frequency � must be rational, and is relatedto the period ( by � � '�( for some integer '. Thefollowing sets are used in this definition: � � � ' � �(�' � (�'�' �(�'� �(� , � � � �'� �(��'� (��'��'� (��'� �(� , and� � � � �(��(� � (� �(� .

3

Signal FS

� � � Reals�

���� � �����

where �� �� �.

� � � Integers�

�� �

�� if � � �� otherwise

� � � Reals�

���� � ��������

where �� �� �.

� � � Integers�

�� �

���� if � � �� otherwise

� � � Reals�

���� � ��������

where �� �� �.

� � � Integers�

�� �

����� if � � ������ if � � ��� otherwise

� � � Reals�

���� � �

� � � Integers�

�� �

�� if � � �� otherwise

Square wave:

� � � ��� ���

���� �

�� if � � � or � � �� �� otherwise

� � � Integers�

�� �������� �

����

Impulse train:

� � � Reals�

���� �

���

��� ���

� � � Integers�

�� � ���

Fourier series coefficients of periodic signals. In all cases, � � ����(, where ( is the period.

4

Signal DTFT

� � � Integers� ���� � ��� � � � Reals� ���� � �

� � � Integers����� � �����

� � � Reals� ���� � ����

� � � Integers� ���� � ! � � � Reals�

���� � ��!�

��

�� � ����

� � � Integers�

���� � ��"���� � � �

� � � Reals�

���� ��

�� ����

� � � Integers�

���� �

�� if � �� otherwise

� � � Reals�

���� ������� � � ���

�������

� � � Integers�

���� ����# ��

��� � � # � �

� � � ���� ���

���� �

�� if � #� otherwise

Discrete time Fourier transforms of key signals. The function 0is the unit step.

5

Signal CTFT

� � � Reals� ���� � ��� � � � Reals� ���� � �

� � � Reals� ���� � ��� ��� � �Reals

� � � Reals� ���� � ����

� � � Reals� ���� � ! � � � Reals� ���� � ��!���

� � � Reals�

���� � �"���� � � � � �

� � � Reals�

���� ��

$� � ����

� � � Reals�

���� �

���� if � �� otherwise

� � � Reals�

���� ��� ������

��

� � � Reals�

���� �������� �

�����

� � � Reals�

���� �

�� if � ���� otherwise

Continuous time Fourier transforms of key signals. The function0 is the unit step.

6

Time domain Frequency domain

� � � Integers� ���� is real � � � Integers� ��� � � ��

� � � Integers� ���� � ������ � � � Integers� ��� is real

� � � Integers� &��� � ������ � � � Integers� ' �� � ������� �

� � � Integers� &��� � ���������where �� � ���, for some � � Integers

� � � Integers� ' �� � � �

��

� � � Integers�&��� � �����������

where �� � ���, for some � � Integers

� � � Integers�' �� � �� �

�� � � ���� ���

� � � Integers�&��� � �����������

where �� � ���, for some � � Integers

� � � Integers�' �� � �� �

�� �� ���� ����

� � � Integers�&��� � ����� � ()���

� � � Integers�' �� � �� �

� � (# ��

� � � Integers� &��� � ����� � � � Integers� '� � ���

Properties of the DFT. All time-domain signals are assumed tobe periodic with period (, and fundamental frequency �� � ���(.

7

Time domain Frequency domain

� � � Reals� ���� is real � � � Integers� �� � ���

� � � Reals� ���� � ������ � � � Integers� �� is real

� � � Reals� &��� � ���� � � � � � Integers� '� � ��������

� � � Reals� &��� � ��������where �� � ���, for some � � Integers

� � � Integers� '� � ���

� � � Reals�&��� � �����������

where �� � ���, for some � � Integers

� � � Integers�'� � ���� � ���� ���

� � � Reals�&��� � �����������

where �� � ���, for some � � Integers

� � � Integers�'� � ���� ����� ����

� � � Reals�&��� � ����� � ()���

� � � Integers�'� � ��� � (#�

� � � Reals� &��� � ����� � � � Integers� '� � ���

Properties of the Fourier series. All time-domain signals areassumed to be periodic with period (, and fundamental frequency �� ����(.

8

Time domain Frequency domain

� � � Integers� ���� is real � � � Reals� ���� � ������

� � � Integers� ���� � ������ � � � Reals� ���� is real

� � � Integers� &��� � ������ � � � Reals� ' ��� � ��������

� � � Integers� &��� � ��������� � � � Reals� ' ��� � ��� � ���

� � � Integers�&��� � �����������

� � � Reals�' ��� � ���� � ��� � ��� � ������

� � � Integers�&��� � �����������

� � � Reals�' ��� � ���� � ������� � �������

� � � Integers����� � ������ � (�����

� � � Reals����� � ������ � (�����

� � � Integers� &��� � �* � ����� � � � Reals� ' ��� � +�������

� � � Integers� &��� � �������� � � � Reals�

' ��� � ���

��"�

����, �� � ��%�

� � � Integers�

&��� �

������� � is a multiple of �� otherwise

� � � Reals, Y(w)=X(Nw)�' ��� � �����

Properties of the DTFT.

9

Time domain Frequency domain

� � � Reals� ���� is real � � � Reals� ���� � ������

� � � Reals� ���� � ������ � � � Reals� ���� is real

� � � Reals� &��� � ���� � � � � � Reals� ' ��� � ��������

� � � Reals� &��� � �������� � � � Reals� ' ��� � ��� � ���

� � � Reals�&��� � �����������

� � � Reals�' ��� � ���� � ��� � ��� � ������

� � � Integers�&��� � �����������

� � � Reals�' ��� � ���� � ������� � �������

� � � Reals����� � ������ � (�����

� � � Reals����� � ������ � (�����

� � � Reals� &��� � �* � ����� � � � Reals� ' ��� � +�������

� � � Reals� &��� � �������� � � � Reals�

' ��� � ���

"

����, �� ���%�

� � � Reals�&��� � �����

� � � Reals, Y(w)=(1/|a|) X(w/a)�' ��� � ������

Properties of the CTFT.

10

(continuação do formulário)

11

(fim do formulário)

12