Digital Signal Processing Lecture 2 PDF

Misr Higher Institute For Commerce & Computers Computer Sciences Department Digital Signal Processing DSP lecture 2 DR. Lobna Mohamed Abouelmagd  A signal is a function that conveys information about the behavior or attributes of some phenomenon....

Misr Higher Institute For Commerce & Computers Computer Sciences Department Digital Signal Processing DSP lecture 2 DR. Lobna Mohamed Abouelmagd  A signal is a function that conveys information about the behavior or attributes of some phenomenon. Signals can be continuous or discrete.  - Continuous-time signals are defined for every value of time t.  For example, x(t) = sin(2π f t) is a continuous-time signal where t is a continuous variable.  - Discrete-time signals are defined only at discrete points in time. For example, x[n] = sin(2π f nT) where n is an integer, and T is the sampling period. REPRESENTATION OF DISCRETE-TIME SIGNALS  Discrete-time signals are signals which are defined only at discrete instants of time. For those signals, the amplitude between the two time instants is just not defined. For discrete- time signal the independent variable is time n, and it is represented by x (n).  There are following four ways of representing discrete- time signals:  1. Graphical representation  2. Functional representation  3. Tabular representation  4. Sequence representation Graphical Representation  Consider a single x (n) with values  X (-2) = -3, x(-1) = 2, x(0) = 0, x(1) = 3, x(2) = 1 and x(3) = 2  This discrete-time single can be represented graphically as Functional Representation Tabular Representation Sequence Representation Time Domain and Frequency Domain  Time Domain  The time domain represents a signal as it varies over time. When we observe or measure a signal directly, we usually see it in the time domain. This is the most intuitive way to understand a signal’s behavior, especially for signals that change over time, such as: audio signals or sensor readings.  Representation In the time domain, a signal x(t) for continuous time or x[n] for discrete time is represented as a function of time. The amplitude of the signal is plotted against time. x(t) = Asin(2π f t + ϕ)  where:  A is the amplitude, representing the signal’s maximum value.  f is the frequency, indicating how many cycles the signal completes per second.  ϕ is the phase, determining the signal’s horizontal shift.  Example: Sine Wave A simple example of a time-domain signal is a sine wave. It can be represented as: x(t) = sin(2π50t)  This equation describes a sine wave with a frequency of 50 Hz. Frequency Domain  The frequency domain represents a signal in terms of its frequency components rather than time. While the time domain shows how a signal changes over time, the frequency domain shows how much of the signal lies within each given frequency band over a range of frequencies.  Representation In the frequency domain, a signal X( f ) for continuous frequency or X[k] for discrete frequency is represented as a function of frequency. The amplitude (or magnitude) and phase of each frequency component are plotted against frequency. X( f ) = Magnitude at frequency f  The ADC unit  samples the analog signal,  quantizes the sampled signal,  encodes the quantized signal levels to the digital signal. Sampling of Continuous Signal An analog (continuous-time) signal (solid line) defined at every point over the time axis (horizontal line) and amplitude axis (vertical line). It is impossible to digitize an infinite number of points infinite amount of infinite amount processing power of memory for computations  Sampling can solve such a problem by taking samples at the fixed time interval  where the time T represents the sampling interval or sampling period in seconds.  each sample maintains its voltage level during the sampling interval T to give the ADC enough time to convert it. This process is called sample and hold. 14 samples at their sampling time instants are plotted For a given sampling interval T, which is defined as the time span between two sample points, the sampling rate is therefore given by  examples  if a sampling period is T = 125 microseconds  the sampling rate is determined as  fs = 1/125 μs = 8,000 samples per second (Hz).  After the analog signal is sampled --- obtain the sampled signal whose amplitude values are taken at the sampling instants --- the processor is able to handle the sample points.  Next, we have to ensure that samples are collected at a rate high enough that the original analog signal can be reconstructed or recovered later.  we are looking for a minimum sampling rate to acquire a complete reconstruction of the analog signal from its sampled version.  If an analog signal is not appropriately sampled, aliasing will occur, which causes unwanted signals in the desired frequency band  The sampling theorem  The sampling theorem guarantees that an analog signal can be in theory perfectly recovered as long as the sampling rate is at least twice as large as the highest-frequency component of the analog signal to be sampled. where fmax is the maximum-frequency component of the analog signal to be sampled. For example, to sample a speech signal containing frequencies up to 4 kHz, the minimum sampling rate is chosen to be at least 8 kHz, or 8,000 samples per second; to sample an audio signal possessing frequencies up to 20 kHz, at least 40,000 samples per second, or 40 kHz, of the audio signal are required.  The sampling theorem is satisfied, 2fmax = 80 Hz < fs. the condition of the sampling theorem is not satisfied sample points is T = 0.01 second, thus the sampling rate is fs = 100 Hz. the sampling theorem in frequency domain 