بسم الله الرحمن الرحيم Lecture (3) Frequency Domain Concepts Dr. Iman Abuel Maaly Lecture (3) Frequency Domain Concepts Dr. Iman Abuel Maaly University of Khartoum Department of Electrical and Electronic Engineering Fourth Year ( ) Digital Signal processing معالجة الإشارة الرقمية Digital Signal processing معالجة الإشارة الرقمية

ContentContent Frequency Domain Concepts  Continuous-time sinusoidal signals  Discrete-time sinusoidal signals  Harmonically Related Complex Exponentials

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4 The Concept of Frequency in Continuous- time signals The concept of frequency is directly related to the concept of time. A simple harmonic oscillation: Signal parameters: A : Amplitude Ω : Frequency in radians/sec θ : phase in radians Ω = 2π F F : frequency in cycles/sec or Hz. T=1/F is the period of the signal

The Concept of Frequency in Continuous- time signals

The Analog signal is characterized by the following characteristics: 6

For F= 0, T p = ∞

8 Continuous time sinusoidal signals By definition frequency is an inherently positive physical quantity. However, only for mathematical convenience, we need to introduce negative frequencies.

As time progresses the phasors rotate in opposite directions with angular frequencies : ± Ω radians/sec Continuous time sinusoidal signals

The Concept of Frequency in Continuous- time signals Representation of a cosine function by phasors

Discrete-time sinusoidal signals Signal parameters: n : integer variable A : Amplitude ω : Frequency in radians/sample (ω = 2π f) θ : phase in radians f : frequency in cycles/sample.(f=1/N) N : is the period

Example Example of a discrete-time sinusoidal signal: (ω =π/6 radians per sample (f =1/12 cycles per sample) and θ = π/3 ) n x(n)

Discrete-time sinusoidal signals (cont) 1 Properties B1. A Discrete time sinusoid is periodic only if its frequency f is a rational number B1. A Discrete time sinusoid is periodic only if its frequency f is a rational number.

Proof: By definition, a discrete time signal is periodic if and only if x(N+n) =x(n) f 0 is the fundamental frequency The relationship is true if there exists an integer k such that

Discrete-time sinusoidal signals (cont) A discrete-time signal is periodic, if its frequency f 0 can be expressed as the ratio of two integers. (i.e., f 0 is rational.) integer

Discrete-time sinusoidal signals (cont) Any small change in frequency can result a large change in the period:

Discrete-time sinusoidal signals (cont) Properties B2. Discrete-time sinusoids whose frequencies are separated by an integer multiple of 2 π are identical.

Proof Consider the sinusoid As a result, all sinusoidal sequences Where are indistinguishable (i.e., identical) Discrete-time Sinusoidal Signals (cont)

On the other hand, Sequences of any two sinusoids with frequencies in the range are distinct (unique) Consequently, discrete time sinusoidal signals with frequencies

21 Any sequence resulting from a sinusoid with frequencies is identical to a sequence obtained from a sinusoidal with frequencies Because of this similarity, we call the sinusoid having the frequency an alias of a corresponding sinusoid with frequency. Discrete-time sinusoidal signals (cont)

22 NoticeNotice The difference between discrete time sinusoids and continuous time sinusoids, is that the continuous time signals result in distinct signals for Ω or F in the range Discrete-time sinusoidal signals (cont)

Discrete-time sinusoidal signals (cont) Properties B3. The highest rate of oscillation in a discrete-time sinusoids is attained when ω = π ( or ω = -π) or equivalently f = ½ ( or f = - ½ )

Discrete-time sinusoidal signals (cont) When the frequency varies from 0 to π. We take values of ω 0 =0, π/8, π/4, π/2, π corresponding to f = 0, 1/16, 1/8, ¼, ½. Which result in periodic sequences having periods N=∞, 16, 8, 4, 2 as depicted in the Figure below. 24

Harmonically Related Complex Exponentials These are sets of periodic complex exponentials with fundamental frequencies that are multiples of a single positive frequency

Text Book: John G. Proakis, and D. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications, This lecture covered the following:

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