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I want to explain this periodicity issue by giving some examples. A sine (or cosine) wave as everyone knows is formed by only one frequency (single period) component. So in the frequency domain it will be only one point showing the amplitude of the signal at that frequency. Shortly, the sine wave is continiuous and periodic but its spectrum is aperiodic and discrete.
With inverse Fourier transform you can generate the original signal easily by the parametres known by the spectrum: amplitude and frequency.
If you superpose a series of sines(or cosines) you have a periodic signal but the spectrum will be made of discrete points representing each of the component's amplitude and frequency points.
In fact, if you are a beginner, if you want to understand the signal characteristics deeply.
you can use matlab to simulate them, so you can observe their output results.
through this method, you can find out many results that don't find in this book.
The diagrams in DSP book by Proakis-Manolakis are perfect...
If you want to understand the underlying funda, you must try to visualise how sinusoids undergo superposition to form a particular signal, periodic or aperiodic. First try to visualise for periodic and then proceed towards aperiodic. As the signal becomes aperiodic, when you try to "construct" it using periodic sinusoids, you need to have infinite signals upto infinite frequency (there are exceptions!). Hope this will help you realise the beauty of the entire transform concept.
Fourier series is used to represent a periodic signal as a linear comb. of harmincally related exponentials (from sophomore level course). As a result of periodicity these signals possess a line spectre with equidistant lines. Line spacing = fundamental frequency (1/T where T is the fundamental period). If we allow the period to increase without limit then the line spacing will tend to zero. In the limit when the period becomes infinite the signal becomes aperiodic and the spectrum becomes continuous.