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The vast majority of the QPM schemes use the multiple-domain ferroelectric crystals of lithium niobate (PPLN). We have measured temporal and spectral characteristics of the THz wave radiation generated under QPM optical rectification of the femtosecond laser pulses in PPLN crystal samples of two different types. The first-type QPM samples were fabricated by applying the standard electrical poling technique to thin slabs cut from the LiNbO3 crystals in the post-growth period. The second-type QPM samples were cut from the Mg:Y:LiNbO3 crystals obtained directly within the off-center Czochralski growth procedure. Due to the technical features of the post-growth method, the input and output windows of the post-growth crystals are limited along the polar Z-axis of the crystal, usually up to 1-5 THz wavelengths; the windows of the in-growth samples are wider in 5-10 times. Previously, the larger input aperture enabled to realize the QPM electro-optical detection in the in-growth PPLN crystals, while the detection efficiency of the post-growth samples was too low. At present, our study demonstrates that the generation efficiencies of the post-growth samples and the most regular parts of the in-growth crystals are of the same order. Stochastic character of the in-growth samples does not affect appreciably the width of the generated spectrum, but decreases the peak power. A special theoretical and experimental study was done to estimate what kind of the sample aperiodicity appears in the QPM THz waveforms, aperiodicity of the several front-age domains, located within the THz wave absorption length, or aperiodicity of the total domain structure of the sample. We demonstrate, that the domain structure, located within the whole crystal, assembles the THz waveform, detected within the first time-domain intervals. The method of optical rectification is a well-known approach for generation of short THz pulses under femtosecond-pulsed laser pumping. Nevertheless, we show that this method can be extended for the case of generation of nanosecond-scale THz pulses. The experimental implementation of this approach for quasi-continuous THz wave generation using PPLN crystals of the both types, as well as bulk ZnTe crystals, will be discussed.