formulário de sistemas e sinais - …¡rio de sistemas e sinais (leic – alameda – 2007/2008 –...
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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
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InverseCTFT� ContSignals� ContSignals
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FourierSeries�� ContPeriodic� � DiscSignals
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InverseFourierSeries��DiscSignals� ContPeriodic�
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Periodicfrequency
Discretetime
DTFT� DiscSignals� ContPeriodic��
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InverseDTFT� ContPeriodic�� � DiscSignals
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��
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��
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DFT�� DiscPeriodic� � DiscPeriodic�
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�������
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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